De Seriebus Infinitis Tractatus. Pars Prima. Auctore Petro Remundo de Monmort. R. S. S.

II. De Seriebus infinitis Tractatus. Pars Prima. Auctore Petro Remundo de Monmort. R. S. S. Prop. 1. Prob. Invenire summam terminorum quot libuerit Seriei hujus \(a \times a + n \times a + 2n \times \&c. \times a + p - 1n\) \(+ a + n \times a + 2n \times a + 3n \times \&c. \times a + pn\) \(+ a + 2n \times a + 3n \times a + 4n \times \&c. \times a + p + 1n\) \(+ a + 3n \times \&c.\). Ubi est \(n\) differentia data, ram inter Factores continuos, \(a, a + n, a + 2n, \&c.\) ejusdem cujusvis termini, quam inter Factores homologos terminorum diversorum in Serie continuatâ; atque designat \(p\) numerum factorum hujusmodi in quovis termino. Solutio Per \(x\) designetur primus Factorum in ultimo terminorum quorum summam requiritur, atque summam illam erit \(x \times x + n \times \&c. \times x + pn = a - n \times a \times \&c. \times a + p - 1n\) Q.E.I. Ex. 1. Proponatur Series numerorum naturalium \(1 + 2 + 3 + 4 + \&c.\) & invenienda sit summa tot terminorum quot sunt unitates in numero \(z\), qui in hoc casu est etiam ultimus terminorum quorum summam requiritur. In hoc itaque casu sunt \(a = 1, n = 1, p = 1, \&c.\) \(x = z\). Unde fit \(x \times x + n \times \&c. \times x + pn = z \times z + 1\), \(a - n \times a \times \&c. \times a + p - 1n = 0 \times 1\), atque \(p + 1n = 2 \times 1\); adeoque summa quaesita est \(\frac{z \times z + 1}{2}\). Ex. 2. Invenienda sit summa tot terminorum, quot sunt unitates in numero \(z\), Seriei \(1 + 3 + 6 + 10 + \&c.\). Numerorum Triangularium. Numeri \(1, 3, 6, 10, \&c.\) in hac Serie Serie sic scribi possunt $\frac{1 \times 2}{2}, \frac{2 \times 3}{2}, \frac{3 \times 4}{2}, \frac{4 \times 5}{2}, \&c.$ Hoc pacto, seposito divisore dato 2, Series revocatur ad formam Propositionis, existentibus $a = 1$, $n = 1$, &c. $p = 2$. $x = z$. Unde summa Seriei duplicata est $$\frac{x \times x + 1 \times x + 2}{3} = \frac{x \times x + 1 \times x + 2}{3};$$ ad eoque habita ratione divisoris 2, Summa Seriei ipsius est $$\frac{x \times x + 1 \times x + 2}{2 \times 3}, \text{ vel } \frac{x \times x + 1 \times x + 2}{2 \times 3}, \text{ in hoc casu existente } x \text{ eodem ac } z.$$ Ad eundem modum inveniuntur summæ cæterorum numerorum figuratorum, quorum formulæ jam vulgò innotescunt. Ex. 3. Sint $a = 1$, $n = 2$, $p = 3$, ut sit Series proposita $1 \times 3 \times 5 + 3 \times 5 \times 7 + 5 \times 7 \times 9 + \&c.$ In hoc itaque casu formula summæ fit $$\frac{x \times x + 2 \times x + 4 \times x + 6 - 1 - 2 \times 1 \times 3 \times 5}{8} = \frac{4 \times 2}{8}$$ $$\frac{x \times x + 2 \times x + 4 \times x + 6 + 15}{8}.$$ Verbi gratiâ, si quaeratur summa decem terminorum, fit $x = 19$ (nempe terminus decimus in Serie Arithmetice proportionalium, $1, 3, 5, 7, \&c.$) adeoque summa est $$\frac{19 \times 21 \times 23 \times 25 + 15}{8} = 28680.$$ Propositio vero sic demonstratur. Demonstratio. Sit Series quantitatum $A, B, C, D, E, \&c.$ quarum differentiae constituant Seriem $a, b, c, d, \&c.$ (nempe ut sint $a = B - A$, $b = C - B$, $c = D - C$, &c.) Hinc statim colligitur esse $a + b = C - A$, $a + b + c = D - A$, $a + b + c + d = E - A$: & in genere aggregatum quotlibet terminorum Seriei $a, b, c, d, \&c.$ æquale est termino proximè sequenti Seriei $A, B, C, D, E, \&c.$ multiplicato termino primo $A$. Pro $A, B, C, \&c.$ sum terminos $$a - n$$ \[ \frac{a - n \times a \times \text{etc.} \times a + p - 1 \times n}{p + 1 \times n}, \quad \frac{a \times a + n \times \text{etc.} \times a + p \times n}{p + 1 \times n} \] \[ \frac{a + n \times a + 2 \times n \times \text{etc.} \times a + p + 1 \times n}{p + 1 \times n}, \text{etc. hoc est, valores successivos ipsius} \] \[ \frac{x \times x + n \times \text{etc.} \times x - p \times n}{p + 1 \times n}; \text{et com-} \] \[ \text{orum differentiæ, pro } a, b, c, d, \text{ etc. sumendæ, erunt} \] \[ a \times a + n \times \text{etc.} \times a + p - 1 \times n, a + n \times a + 2 \times n \times \text{etc.} \times a + p \times n, \text{ etc. qui sunt ipsissimi termini Seriei propositæ. Sed} \] \[ \text{comparando has Series, si terminus aliquis Seriei posterioris sit } x \times x + n \times \text{etc.} \times x + p - 1 \times n, \text{ constat terminum uno ulteriorem in Serie priori fore} \] \[ x \times x + n \times \text{etc.} \times x + p \times n. \text{ Summa itaque Seriei posterioris usque terminum } x \times x + n \times \text{etc.} \times x + p - 1 \times n \text{ inclusivè est} \] \[ x \times x + n \times \text{etc.} \times x + p \times n - a - n \times a \times \text{etc.} \times a + p - 1 \times n \] \[ Q.E.D. \] **Scholium 1.** In hâc propositione continetur particula quædam Methodi incrementorum, de quâ ante biennium librum edidit D. Brook Taylor Soc. Reg. Lond. Secr. mihi amicitiâ conjunctissimus. Librum ipsum adeat qui de eâ methodo plura scire velit: ad institutum nostrum sufficit observare quanta intersit affinitas inter Methodum hanc & Methodum Fluxionum seu differentiale Nam ut in Methodo differentiali, ad inveniendum differentiale ipsius \( x \) dignitatis \( x^n \), unum latus \( x \) convertendum est in differentiam \( dx \); & ortum ducendum est in dignitatis Indicem \( m \), ut sit \( m \times dx \times x^{m-1} \) differentiale quaesturum; sic in Methodo Incrementorum Ad inveniendum Incrementum facti hujusmodi \( x \times x + n \times x + 2 \times n \), ubi factores \( x, x + n, x + 2 \times n, \) \( x + 2n \), sunt in progressionem Arithmeticae, cujus differentia communis est ipsius \( x \) Incrementum datum \( n \). Factorum minus \( x \) convertendus est in Incrementum, & ortum ducendum est in numerum Factorum, ut sit \( 3n \times x + n \times x + 2n \) Incrementum quasitum, numero Factorum in casu exposito existente 3. Sic etiam ipsius \( x \times x + n \) Incrementum fit \( 2n \times x + n \). 2. Incrementa etiam Reciprocorum hujusmodi Factorum inveniuntur per eandem regulam; hoc nempe observato, quod cum sit Divisio contrarium Multiplicationis, vice ablationis minimi Factorum, sit jam addendus alius factor adhuc uno Incremento major; item quod Factorum numerus sit scribendus cum signo negativo. Hoc pacto ipsius \( \frac{1}{x} \) Incrementum fit \( \frac{-1 \times n}{x \times x + n} \); ipsius \( \frac{1}{x \times x + n} \) Incrementum fit \( \frac{-2 \times n}{x \times x + n \times x + 2n} \); & sic de aliis hujusmodi. Hoc facile probatur sumendo differentias inter Integralium valores duos continuos. 3. Insistendo vestigiis Methodi directae, hinc colliguntur praecepta Methodi inversae, quibus inveniuntur Integralia Incrementorum oblatorum. Applicetur enim Incrementum oblatum ad lateris Incrementum datum; addatur Factor adhuc uno Incremento minor, & applicetur ortum ad numerum Factorum hic autorum. Sic e.g. oblatum Incremento \( n \times x \times x + n \times x + 2n \) fit primò \( x \times x + n \times x + 2n \); deinde \( x - n \times x \times x + n \times x + 2n \), addito Factori \( x - n \); denique \( x - n \times x \times x + n \times x + 2n \), quod est Integrale quassitum. Hoc quidem ubi Factori sunt Multiplicantes; Ubi vero Factori occupant locum diviforis, mutatis mutandis, regula hæc est, Applicetur Incrementum oblatum ad lateris incrementum datum; rejiciatur Factorum Factorum maximus, & applicetur ortum ad numerum Factorum relictorum cum signo negativo. Exempli gratia oblatu Incremento \( \frac{n}{x \times x + n \times x + 2n} \), sit primo \( \frac{1}{x \times x + n \times x + 2n} \), deinde \( \frac{1}{x \times x + n} \), denique \( \frac{1}{-2 \times x \times x + n} \), seu \( \frac{1}{2 \times x \times x + n} \), quod est Integrale quaestium. 4. In casu hoc novissimo Integrale inventum, cum signo contrario, æquale est summæ omnium Incrementorum in Serie in infinitum continuatâ; v.g. est \( \frac{1}{2 \times x \times x + n} \) \( = \frac{n}{x \times x + n \times x + 2n} + \frac{n}{x + n \times x + 2n \times x + 3n} \) \( + \frac{n}{x + 2n \times x + 3n \times x + 4n} + \&c. \) Nam in hoc casu, facto \( x \) tandem infinito, evanescit \( \frac{1}{2 \times x \times x + n} \), hoc est, ultimus terminorum \( A, B, C; \&c. \) fit nihil; & ob contrarietatem signorum Integralis & Incrementi, vice \( -A \) exprimitur aggregatum per \( +A \). Lemma 1. Per \( X \) designetur terminus quilibet in Serie quâvis numerorum \( M, N, O, P, \&c.; \) per \( x \) designetur locus termini istius \( X \) in Serie illâ (v.g. ut sit \( x = 1 \), quando designat \( X \) terminum primum \( M \), sit \( x = 2 \), quando designat \( X \) terminum secundum \( N \), & sic de cæteris) & sint terminorum \( M, N, O, P \) prima differentiarum primarum \( b, c \) prima differentiarum secundarum, \( d \) prima tertiarum, \( e \) prima quartarum, & sic porrò. Tum erit \[ F f f f f f \quad X = M \] \[ X = M + b \times \frac{x-1}{1} + c \times \frac{x-1}{1} \times \frac{x-2}{2} + d \times \frac{x-1}{1} \times \frac{x-2}{2} \times \frac{x-3}{3} \times \frac{x-4}{4} + \&c. \] Sequitur hoc ex tabula aequationum p.e.g. 66. tractatus nostri Essay d'Analyse, &c. Lemma 2. Iisdem positis, per \( z \) designetur terminus quilibet in Serie Arithmetice proportionalium \( a, a+n, a+2n, \&c. \) &c. sit jam \[ X = A + Bz + Cz \times z + n + Dz \times z + n \times z + 2n + Ez \times z + n \times z + 2n \times z + 3n + \&c. \] Tum illorum \( A, B, C, D, E, \&c. \) valores erunt. \[ A = M + b \times \frac{-a}{n} + c \times \frac{-a-n}{n} \times \frac{-a-2n}{2n} + \] \[ + d \times \frac{-a}{n} \times \frac{-a-n}{2n} \times \frac{-a-2n}{3n} + \] \[ + e \times \frac{-a}{n} \times \frac{-a-n}{2n} \times \frac{-a-2n}{3n} \times \frac{-a-3n}{4n} + \&c. \] \[ B = \frac{1}{n} \times b + c \times \frac{-a-n}{n} + d \times \frac{-a-n}{n} \times \frac{-a-2n}{2n} + \] \[ + e \times \frac{-a-n}{n} \times \frac{-a-2n}{2n} \times \frac{-a-3n}{3n} + \&c. \] \[ C = \frac{1}{n} \times \frac{1}{2n} \times c + d \times \frac{-a-2n}{n} + e \times \frac{-a-2n}{n} \times \frac{-a-3n}{2n} + \&c. \] \[ D = \frac{1}{n} \times \frac{1}{2n} \times \frac{1}{3n} d + e \times \frac{-a-3n}{n} + \&c. \] \[ E = \frac{1}{n} \times \frac{1}{2n} \times \frac{1}{3n} \times \frac{1}{4n} e + \&c. \] Ordo Ordo formandi coefficientes ipsorum \( b, c, d, e, \ldots \) in his valoribus, per se ei satis manifestius. Demonstratio. Quoniam per \( x \) & \( z \) designantur termini correspondentes progressionum Arithmeticarum 1, 2, 3, 4, \&c. &c. \( a + n, a + 2n, a + 3n, \ldots \), indicabit \( x - 1 \) numerum differentiarum \( n \) qui in \( z \) continetur, ut sit \[ z = a + x - 1n. \quad \text{Hinc fit} \quad x - 1 = \frac{z - a}{n}, \quad x - 2 = \frac{z - n - a}{n}, \quad x - 3 = \frac{z - 2n - a}{n}, \ldots \text{etc.} \] Substituendo itaque hos valores \( x - 1, x - 2, x - 3, \ldots \) in Serie Lemmatis praecedentis, & termi is in ordinem redactis, prodeunt ipsorum \( A, B, C, \ldots \) valores exhibiti. Cor. Ubi \( a = n \), prodeunt \( A, B, C, D, \ldots \) per formulas simpliciores, nempe \[ A = m - b + c - d + e \ldots \] \[ B = \frac{1}{n} \times b - 2c + 3d - 4e \ldots \] \[ C = \frac{1}{n} \times \frac{1}{2n} \times c - 3d + 6e \ldots \] \[ D = \frac{1}{n} \times \frac{1}{2n} \times \frac{1}{3n} \times d + 4e \ldots \] Lemma 3. Symbolis \( X \) & \( x \) eodem modo interpretatis ac in Lemmate primo, sint \( q, r, s, t, u, \ldots \) generatores Trianguli Arithmetici cujus lineam transversam, occupat Series \( M, N, O, P, Q, \ldots \) in ordine nempe inverso, ut sit \( q (= M) \) generator ultimus, \( r \) penultimus, \( s \) antepenultimus, \&c. sic porrò. Tum erit \[ X = q + r \times \frac{x - 1}{1} + s \times \frac{x - 1}{1} \times \frac{x}{2} + t \times \frac{x - 1}{1} \times \frac{x}{2} \times \frac{x + 1}{3} + \ldots \] Constat Constat ex contemplatione ipsius Trianguli Arithmetici, quam exhibuimus pag. 63 tractatus Essay d'Analyse, &c. ubi idem fusius explicatur. Lemma 4. Iisdem positis, & Symbolo z eodem modo interpretato ac in Lem. 2. si sit $X = A + Bz + Cz \times z + n + \&c.$ ut in Lem. 2. erunt coefficientium $A, B, C, D, \&c.$ valores. $$A = q + r \times \frac{-a}{n} + s \times \frac{-a}{n} \times \frac{-a + n}{2n} + t \times \frac{-a}{n} \times \frac{-a + n}{2n} \times \frac{-a + 2n}{3n} + \&c.$$ $$B = \frac{1}{n} \times r + s \times \frac{-a}{n} + t \times \frac{-a}{a} \times \frac{-a + n}{2n} + \&c.$$ $$C = \frac{1}{n} \times \frac{1}{2n} \times s + t \times \frac{-a}{n} + \&c.$$ $$D = \frac{1}{n} \times \frac{1}{2n} \times \frac{1}{3n} \times t + \&c.$$ Ordo coefficientium in his valoribus est manifestus, & demonstratur Lemma ad modum Lemmatis 2. Cor. 1. Ubi $a = n$, coefficientes, $A, B, C, D, \&c.$ produent per formulas simpliciores, nempe $$A = q - r,$$ $$C = \frac{1}{n} \times \frac{1}{2n} \times s - t$$ $$B = \frac{1}{n} \times r - s,$$ $$D = \frac{1}{n} \times \frac{1}{2n} \times \frac{1}{3n} t - u$$ Cor. 2. Unde si generatorum $q, r, s, t, u, \&c.$ aliquot sint inter se æquales, exhibebitur $X$ per formulam simpliciorem, evanescentibus aliquot coefficientium $A, B, C, D, \&c.$ Sic Sic exempli gratiâ, propositâ Serie numerorum 4, 69, 530, 2676, 10350, &c. qui constituunt lineam decimam transversam in Triangulo Arithmetico cujus generatores tres priores sunt 54, — 18, 5, & septem posteriores sunt æquales 4; existente \(a = 1 = n\), Terminus \(X\) exhibetur per formulam quatuor tantum terminorum. \[ -\frac{z}{1} \cdot \frac{z + 1}{2} \cdot \frac{z + 2}{3} \cdot \&c. \times \frac{z + 6}{7} + 23 \frac{z}{1} \cdot \frac{z + 1}{2} \cdot \&c. \] \[ \times \frac{z + 6}{7} - 72 \frac{z}{1} \cdot \frac{z + 1}{2} \cdot \&c. \times \frac{z + 7}{8} + 54 \frac{z}{1} \cdot \frac{z + 1}{2} \cdot \&c. \] \[ \times \frac{z + 8}{9}. evanescentibus coefficientibus sex primis A, B, C, D, E, F. \] **Prop. II. Prob.** Invenire summam quotlibet terminorum Seriei \[ \frac{M}{a \times a + n \times \&c. \times a + p - 1n} + \frac{N}{a + n \times \&c. \times a + pn} \] \[ + \frac{O}{a + 2n \times \&c. \times a + p + 1n} + \&c. ubi numeratores M, N, O, \&c. constituunt Seriem quamlibet terminorum, quorum differentiae, vel primae, vel secundae, vel aliæ quædam dantur; vel quod perinde est, qui constituunt lineam quamvis transversam in dato quovis triangulo Arithmetico; Denominatores autem constituunt Seriem in Prop. I. exhibitam. *Solutio.* Per \(X\) designetur primus factorum \(a, a + n, a + 2n, \&c.\) in denominatore ejusdem termini, ut sint \(X\) & \(z\) iudem ac in Lemm: præmissis, adeoque designetur terminus quilibet Seriei per \[ \frac{X}{z \times z + n \times \&c. \times z + p - n} \] Per Lem. 2, vel per Lem. 4. (prout magis commodum G g g g g videatur videatur vel differentias, vel generatores trianguli Arithmetici adhibere,) resolvatur $X$ in Multinomium $A + B \times z + Cz \times z + n + Dz \times z + n \times z + 2n + \&c.$ Hoc pacto (terminis multinomii ad denominatorum $z \times z + n \times \&c. \times z + p - n$, applicatis) terminus quilibet Seriei revocabitur ad formulam $$\frac{A}{z \times z + n \times \&c. \times z + p - 1n} + \frac{B}{z + n \times \&c. \times z + p - 1n} + \frac{C}{z + 2n \times \&c. \times z + p - 1n} + \&c.$$ Unde (per Scholium 4 Prop. I.) aggregatum totius Seriei, à termino $$\frac{X}{z \times z + n \times \&c. \times z + p - 1n}$$ inclusi, ve in infinitum continuatæ, est $$\frac{A}{p - 1 \times n \times z \times z + n \times \&c. \times z + p - 2n} + \frac{B}{p - 2 \times n \times z + n \times \&c. \times z + p - 2n} + \frac{C}{p - 3 \times n \times z + 2n \times \&c. \times z + p - 2n} + \&c.$$ re si dematur hoc aggregatum ab ejusdem aggregati valore quando $z = a$, residuum erit summa omnium terminorum ante terminum $\frac{X}{z + \&c.}$, hoc est, tot terminorum quot sunt unitates in $\frac{z - a}{n}$. Q.E.I. Ex. 1. Sit primum exemplum in Serie $\frac{5}{3 \cdot 5 \cdot 7 \cdot 9 \cdot 11 \cdot 13}$. \[ \frac{41}{5 \cdot 7 \cdot 9 \cdot 11 \cdot 13 \cdot 15} + \frac{131}{7 \cdot 9 \cdot 11 \cdot 13 \cdot 15 \cdot 17} + \frac{275}{9 \cdot 11 \cdot 13 \cdot 15 \cdot 17 \cdot 19} + \frac{473}{11 \cdot 13 \cdot 15 \cdot 17 \cdot 19 \cdot 21} + \text{etc.} \text{Sunt hic } a = 3, n = 2, t = 6, M = 5, \&c. \text{capiendo differentias numeratorum inveniuntur } b = 36, c = 54, d = 0 = e = \text{etc.} \text{Hinc in Lemmate secundo sunt } A = 5 + 36 \times \frac{3}{2} + 54 \times \frac{3}{2} \times \frac{5}{4} = \frac{299}{4}, B = \frac{1}{2} \times 36 + 54 \times \frac{5}{2} = \frac{-99}{2}, C = \frac{1}{2} \times \frac{1}{4} \times 54 = \frac{27}{4}, D = 0 = E = \text{etc.} \text{Summa itaque totius Seriei est} \frac{209}{4 \times 5 \times 2 \times 3 \cdot 5 \cdot 7 \cdot 9 \cdot 11} + \frac{-99}{2 \times 4 \times 2 \times 5 \cdot 7 \cdot 9 \cdot 11} + \frac{27}{4 \times 3 \times 2 \times 7 \cdot 9 \cdot 11} = \frac{283}{80 \times 3 \cdot 5 \cdot 7 \cdot 9 \cdot 11}, \text{ atque summa terminorum numero } \frac{z - 3}{2} (\equiv \frac{z - a}{n}) \text{ est} \frac{283}{80 \times 3 \cdot 5 \cdot 7 \cdot 9 \cdot 11} - \frac{209}{40 \times z \cdot z + 2 \cdot z + 4 \cdot z + 6 \cdot z + 8} + \frac{99}{16 \times z + 2 \times z + 4 \cdot z + 6 \cdot z + 8} - \frac{27}{24 \times z + 4 \cdot z + 6 \cdot z + 8} \text{Quarantur v.g. octo termini; tum existente } \frac{z - 3}{2} = 8 \text{ fit } z = 19, \text{ quo valore in formulà adhibito, prodit summa } \frac{155891}{2 \cdot 3 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 7 \cdot 11 \cdot 19 \cdot 23}. \text{Iudem Numeratores occupant lineam tertiam transversam in Triangulo Arithmetico} 54 \cdot 54 \cdot 54 \cdot 54 \cdot 54 \cdot 54 \cdot \text{etc.} - 18 \cdot 36 \cdot 90 \cdot 144 \cdot 198 \cdot \text{etc.} 5 \cdot 41 \cdot 131 \cdot 275 \cdot \text{etc.} Unde Unde in formula Lem. 4. sunt generatores \( q = 5 \), \( r = -18 \); \( s = 54 \), \( t = 0 \) &c. & prodeunt coefficientes \( A = 5 - 18 \times \frac{-3}{2} + 54 \times \frac{-3}{2} \times \frac{-3}{4} = \frac{209}{4} \), \( B = \frac{1}{2} \times -18 + 54 \times \frac{-3}{2} = \frac{-99}{2} \), \( C = \frac{1}{2} \times \frac{1}{4} \times 54 = \frac{27}{4} \), \( D = 0 = E = \&c. \) idem ac supra. Ex. 2. Sit Series \[ \frac{4}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot 10 \cdot 11} + \frac{69}{2 \cdot 3 \cdot \&c. \cdot 12} + \frac{530}{3 \cdot 4 \cdot \&c. \cdot 13} + \frac{2676}{4 \cdot 5 \cdot \&c. \cdot 14} + \frac{10350}{5 \cdot 6 \cdot \&c. \cdot 15} + \&c. \] Ubi sunt \( a = 1 \), \( n = 1 \), \( p = 11 \), atque Numeratores constituent Seriem in Corol. 20. Lem. 4. exhibitam. Applicando itaque valorem \( X \) in Corol. illo ad denominatorem \( z \times z + 1 \times \&c. \times z + 10 \), fit Seriei propositionis Terminus \[ \frac{-1}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \times z + 6 \cdot z + 7 \cdot z + 8 \cdot z + 9 \cdot z + 10} + \frac{23}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \times z + 7 \cdot z + 8 \cdot z + 9 \cdot z + 10} - \frac{72}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \times z + 8 \cdot z + 9 \cdot z + 10} + \frac{54}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \times z + 9 \times z + 10} \] Adeoque per hanc Prop. summa Seriei à termino illo in infinitum continuatæ est \[ \frac{-1}{4 \times 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \times z + 6 \cdot z + 7 \cdot z + 8 \cdot z + 9} + \&c. \] \[ \frac{23}{3 \times 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \times z + 7 \cdot z + 8 \cdot z + 9} \] \[ - \frac{72}{2 \times 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \times z + 8 \cdot z + 9} \] \[ + \frac{54}{1 \times 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \times z + 9} \] Itaque pro \( z \) sumpto 1, fit summa totius Seriei \[ \frac{305}{12 \times 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot 10} \] Et in genere summa terminorum numero \( \frac{z - 1}{1} \), est \[ \frac{305}{12 \times 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot 10} \] \[ + \frac{1}{4 \times 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \times z + 6 \cdot z + 7 \cdot z + 8 \cdot z + 9} \] \[ - \frac{23}{3 \times 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \times z + 7 \cdot z + 8 \cdot z + 9} \] \[ + \frac{72}{2 \times 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \times z + 8 \cdot z + 9} \] \[ - \frac{54}{1 \times 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \times z + 9} \] Scholium 1. In computandis summis hujusmodi Series- rum, calculus plerumque levior est adhibiti generatori- bus trianguli Arithmetici, quam si adhibeantur differen- tiæ. Libet itaque hac occasione ostendere quomodo ex datis differentiis inveniri possunt generatores Trianguli Arithmetici. Sunto itaque \( \omega \) primus Seriei terminus, \( a \) differentia ultima data, \( b \) prima differentiarum penultimarum, \( c \) prima antepenultimarum, & sic porrò \( d, e, \&c. \) atque sint \( t, u, x, y, \&c. \) generatores quasi Trianguli Arith- metici, cujus lineam transversam ordine \( p \) occupet Series \[ H \ h \ h \ h \ h \] proposita. Tum (quod ex contemplatione Trianguli Arithmetici facile constat) sunt \[ a = t \] \[ b = \frac{p - 1}{1} t + u \] \[ c = \frac{p - 1}{1} \times \frac{p - 2}{2} t + \frac{p - 2}{1} u + x \] \[ d = \frac{p - 1}{1} \times \frac{p - 2}{2} \times \frac{p - 3}{3} t + \frac{p - 2}{1} \times \frac{p - 3}{2} u \] \[ + \frac{p - 3}{1} x + y &c. \] Unde colliguntur generatorum valores \[ t = a \] \[ u = b - \frac{p - 1}{1} t \] \[ x = c - \frac{p - 1}{1} \times \frac{p - 2}{2} t - \frac{p - 2}{1} u \] \[ y = d - \frac{p - 1}{1} \times \frac{p - 2}{2} \times \frac{p - 3}{3} t - \frac{p - 2}{1} \times \frac{p - 3}{2} u \] \[ - \frac{p - 3}{1} x &c. \] Ultimus autem generator æqualis est Seriei termino primo \( \omega \). 2. Dnus de Monsoury Abbas Orbacensis mihi amicissimus, & ruri vicinus, postquam cum eo hæc communicaveram, aliam invenit hujus Problematis Solutionem, cujus formulam ob ejus miram simplicitatem hic referre juvat. Itaque in Serie numeratorum sint \( \omega \) terminus primus, \( b \) prima differentiarum primarum, \( c \) prima secundarum, \( d \) prima tertiarum, & sic porro; atque sit termini primi Denominator \( z \times z + n \times &c. \times z + p - 1 n \); Tum summa summa totius Seriei in infinitum continuatæ exhibebitur per formulam \[ \frac{\omega}{n \times p - 1 \times z \times z + n \times \&c. \times z + p - 2 \times n} + \frac{b}{n^2 \times p - 1 \times p - 2 \times z + n \times \&c. \times z + p - 2 \times n} + \frac{c}{n^3 \times p - 1 \times p - 2 \times p - 3 \times z + 2 \times n \times \&c. \times z + p - 2 \times n} + \&c. \] Sit exemplum in Serie \( \frac{5}{3 \cdot 5 \cdot \&c. \cdot 13} + \frac{41}{5 \cdot 7 \cdot \&c. \cdot 15} + \frac{131}{7 \cdot 9 \cdot \&c. \cdot 17} + \frac{275}{9 \cdot 11 \cdot \&c. \cdot 19} + \&c. \) cujus summam jam exhibuimus. In hoc casu sunt \( \omega = 5, b = 36, c = 54, d = o = e = \&c. \). Unde per formulam summæ Seriei integrae fit \( \frac{5}{2 \cdot 5 \times 3 \cdot 5 \cdots 11} + \frac{36}{4 \cdot 5 \cdot 4 \times 5 \cdots 11} + \frac{54}{8 \cdot 5 \cdot 4 \cdot 3 \times 7 \cdots 11} = \frac{283}{80 \times 3 \cdot 5 \cdots 11} \), ut per formulam nostram exhibetur. Si quaeratur summa ejusdem Seriei incipientis à termino decimo \( \frac{2273}{21 \cdots 31} \), in eo casu \( \omega = 2273, b = 522, c = 54, \&c. \) summa esset \( \frac{2273}{2 \cdot 5 \times 21 \cdots 29} + \frac{522}{4 \cdot 5 \cdot 4 \times 23 \cdots 29} + \frac{54}{8 \cdot 5 \cdot 4 \cdot 3 \times 25 \cdots 29} \). Hæc formula est commodissima, & summam exhibet nullo serè negotio, quoties quaeritur summa Seriei integrae, & differentiæ non sunt nimis multæ. Sed ubi plures sunt differentiæ, & quaeritur non Series integra, sed termini tantum initiales aliquammulti, formulæ nostræ sunt commodiores. 3. Quando 3. Quando Serierum termini formantur tantum per Multiplicationem, nec afficiuntur divisoribus variabilibus, summae semper exhiberi possunt per Methodum in Prop. I. traditam, sint licet formulae quantumlibet compositae. Nam possunt semper revocari ad terminos in forma quam postulat Propositio illa. Sic si differentiae ipsorum $z$ & $x$ sint $m$ & $n$, & designetur terminus Seriei per $z x$; hic terminus revocabitur ad formam $\frac{a}{n} z + \frac{n}{m} x z + m$; cujus Integrale datur per Prop. I; nempe quoniam $dx = n$, & $dz = m$, est $dx = dz \times \frac{n}{m}$; unde regrediendo ad integralia fit $x = \frac{n}{m} z + a$ (adjecto invariabili $a$, ut habeatur ratio relationis inter $z$ & $x$ in Seriei termino primo,) quod sic scribi potest $\frac{a}{n} + \frac{n}{m} \times z + m$, ut deinde in $z$ ductum induat formam requisitam. Et ad eundem modum procedere licet in aliis casibus ejusmodi. Sed ubi formulae oblatae divisoribus afficiuntur, eadem ac in Calculo integrali, ut vocant, difficultates occurrent, eadem industria superanda. Nec tamen semper superari possunt. Nam praeterquam quod vix certo scriiri possit quae debet relatio intercedere inter Numeratorem fractionis & Denominatorem, ut formula oblata ad Integrale revocari posset; saepè etiam difficillimum est explorare an adsit jam talis relatio in formulâ istâ, aut si desit, an introduci possit. Quicquid ego in hâc materiae potissimum inveni, continetur in tribus sequentibus propositionibus. Prop. III. Prob. Crescentibus, $z$, $y$, $x$, &c. per differentias datas $n$, $m$, $l$, $o$, &c. invenire valorem numeratoris integri tegri \( N \) ut existente Denominatore \( z \cdot z + n \cdot c \cdot z + p \cdot n \cdot u \cdot u + m \cdot c \cdot u + q \cdot m \cdot y \cdot y + l \cdot c \cdot y + r \cdot l \cdot x \cdot x + e \cdot c \cdot x + s \cdot o \cdot c \cdot c \). Fractio ad Integrale revocari possit. Solutio. Fiat \( N = z + p \cdot n \cdot x + q \cdot m \cdot y + r \cdot l \cdot x \cdot x + s \cdot o \cdot c \cdot c - z \cdot u \cdot y \cdot x \cdot c \cdot c \). atque Integrale crit fractio, cujus Denominator \( z \cdot z + n \cdot c \cdot z + p - i \cdot n \cdot x \cdot u \cdot u + m \cdot c \cdot u + q - i \cdot m \cdot y \cdot y + l \cdot c \cdot y - x - i \cdot l \cdot x + o \cdot c \cdot x + s - i \cdot o \cdot c \cdot c \). existente Numeratore. Differentia enim hujus fractionis est fractio cujus numerator est ipsius \( N \) valor exhibitus, & denominator idem est ac denominator propositus, ut fieri debuit. Ex. 1. Sit denominator propositus \( z \cdot z + 2 \cdot x \cdot u \cdot u + 3 \). In hoc casu sunt \( n = 2, m = 3, p = 1, q = 1 \); adeoque est \( N = z + 2 \cdot x \cdot u + 3 - z \cdot u = 3 \cdot z + 2 \cdot u + 6 \), & per \( \frac{3 \cdot z + 2 \cdot u + 6}{z \cdot z + 2 \cdot x \cdot u \cdot u + 3} \) representatur terminus Seriei summabilis, cujus nempe in infinitum continuatae summa exhibetur per \( \frac{1}{z \cdot u} \). Sint verbi gratia, ipsorum \( z \& u \) primus valor communis 1, atque Series summabilis crit \( \frac{1}{1 \cdot 3 \cdot 1 \cdot 4} + \frac{2}{3 \cdot 5 \cdot 4 \cdot 7} + \frac{3}{5 \cdot 7 \cdot 9 \cdot 10} + \&c \), quippe cujus totius summa est 1. Per \( p \) designetur ordo termini cujusvis in hac Serie, erit \( p = \frac{z - 1 + 2}{2} = \frac{u - 1 + 3}{5} \), adeoque \( z = 2 \cdot p - 1, \& u = 3 \cdot p - 2 \); quibus valori bus pro \( z \& u \) scriptis, designabitur terminus per formulam \( \frac{12 \cdot p - 1}{2 \cdot p - 1 \cdot 2 \cdot p + 1 \cdot 3 \cdot p - 2 \cdot 3 \cdot p + 1} \). Summa autem terminorum omnium ante terminum illum, hoc est terminorum initialium numero \( \frac{z - 1}{2} = p - 1 \), est \[ \begin{array}{cccc} 1 & 1 & 1 & 1 \\ \end{array} \] \[1 - \frac{1}{zn} = \frac{zn-1}{zn}, \text{ hoc est } \frac{6pp - 7p + 1}{2p - 1 \times 3p - 2}. \] Quae pro \( p \) scripto \( p + 1 \), erit \( \frac{p \times 6p + 5}{2p + 1 \times 3p + 1} \). aggregatum tot terminorum initialium quot sunt unitates in \( p \). Ex. 2. Iisdem manentibus \( z, u, n, m \), sit denominator \( z \cdot z + 2 \cdot z + 4 \times u \cdot u + 3 \). Tum per formulam numerator erit \( z + 1 \times u + 3 - zn = 3z + 4n + 12 \), & summa Seriei exhibebitur per formulam \( \frac{1}{z \cdot z + 2 \times n} \). Sit ipsorum \( z \) & \( u \) primus valor communis 1, & hinc elicetur Series \( \frac{19}{1 \cdot 3 \cdot 5 \times 1 \cdot 4} + \frac{37}{3 \cdot 5 \cdot 7 \times 4 \cdot 7} + \frac{55}{5 \cdot 7 \cdot 9 \times 7 \cdot 10} + \&c = \frac{1}{3} \). Scholium. In Seriebus jam expositis eadem ubique est differentia inter factores continuos ejusdem cujusvis termini, ac inter factores homologos terminorum continuorum. In sequentibus exempla quaedam sunt Serierum, quarum summæ in terminis numero finitis exhiberi possunt, quamvis ea regula non observetur. Prop. IV. Prob. Crescente \( z \) per differentias datas \( qn \), invenire numeratorem integrum \( N \), ut ad Integrale revocari possit fractio, cujus Denominator sit ex certo numero \( p \) terminorum \( z, z + n, z + 2n, \&c \). Arithmetice proportionalium in invicem ductorum. Debet autem esse \( q \) numerus integer minor quam factorum numerus \( p \). Solutio. Erit \( N = z + p - 1 \times z + p - 2 \times n \times \&c. \) \( xz + p - qn = z \times z + n \times \&c. \times z + q - 1 \times n \), Integrale. tegrale existente \[ \frac{1}{z \times z + n \times \text{etc.} \times z + p - q - 1} \] Demonstratur ad modum propositionis praecedentis. Sumptis ad libitum \( n, p, q, \) & primo valore \( z, \) hinc oriuntur infinitae Series summabiles, cujusmodi sunt Series tres sequentes. \[ A = \frac{5}{1 \cdot 2 \cdot 3 \cdot 4} + \frac{9}{3 \cdot 4 \cdot 5 \cdot 6} + \frac{13}{5 \cdot 6 \cdot 7 \cdot 8} + \frac{17}{7 \cdot 8 \cdot 9 \cdot 10} \text{ etc.} \] \[ B = \frac{1}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5} + \frac{4}{4 \cdot 5 \cdot 6 \cdot 7 \cdot 8} + \frac{9}{7 \cdot 8 \cdot 9 \cdot 10 \cdot 11} \] \[ + \frac{16}{10 \cdot 11 \cdot 12 \cdot 13 \cdot 14} + \text{etc.} \] \[ C = \frac{1}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5} + \frac{14}{5 \cdot 6 \cdot 7 \cdot 8 \cdot 9} + \frac{55}{9 \cdot 10 \cdot 11 \cdot 12 \cdot 13} \] \[ + \frac{140}{13 \cdot 14 \cdot 15 \cdot 16 \cdot 17} + \text{etc.} \] Has Series jampridem communicavi cum primariis quibusdam Geometris, à quibus minimè contemni videntur. Sic ad me scribit peritissimus Geometra D Nicolaus Bernoulli in epistolâ datâ 25 Julii 1716. "Vous me ferez un extreme plaisir, Monsieur, de me communiquer la Solution de votre problème, Etant donnée une suite des Fractions dont les Numerateurs soient des nombres figurés quelconque, & dont les Denominateurs soient formés du produit d'un nombre egal de Facteurs qui soient en Progression Arithmetique, trouver la somme; & principalement comment vous avez trouvé ces deux formules \[ \frac{p}{24 \times 4^p + 1}, \quad \frac{p \cdot p + 1}{12 \times 3^p + 1 \times 3^p + 2}. \] Hæ formulæ spectant ad Series C & B, designante \( p \) numerum terminorum, quorum summa requiritur. Sic etiam ad me scribit D. Taylor in epistola datâ 22 Aug. 1716. "Ut & quâ ratione incidisti in summationem Serierum à te exhibitarum, præsertim loquor de Serie Serie $\frac{1}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5} + \frac{4}{4 \cdot 5 \cdot 6 \cdot 7 \cdot 8} + \frac{9}{7 \cdot 8 \cdot 9 \cdot 10 \cdot 11} + \&c.$ quae videtur esse altioris indaginis. Sed ut ad exempla jam redeamus. In Serie A sunt $p = 4$, $q = 2$, $n = 1$, primo valore $z$ existente 1. Est itaque $z + 3 \times z + 2 - z \times z + 1 = 2 \times 2z + 3$ formula, unde (rejecto dato numero 2) derivantur numeratores 5, 9, 13, 17, &c. Formula etiam summæ est $\frac{1}{z \times z + 1}$. Quare habita ratione numeri 2, quem ex numeratoribus rejecimus, summa totius Seriei, à termino in quo est $z$ in infinitum continuatæ, exhibetur per formulam $\frac{1}{2 \times z \times z + 1}$; adeoque summa Seriei integræ est $\frac{1}{2 \times 1 \times 2} = \frac{1}{4}$. In Serie B sunt $n = 1$, $p = 5$, $q = 3$, primo valore $z$ existente 1. Est itaque $N = z + 4 \times z + 3 \times z + 2 - z \times z + 1 \times z + z = 6 \times z + 2$. Ipsius autem $z + 2$ valores continui sunt 3, 6, 9, &c. qui quoniam omnes sunt divisibles per 3, ponendo $z + 2 = 3x$, fit $N = 6 \times 3x^2 = 6 \times 9x^2 = 54x^2$, ipsius $x$ valoribus continuis existentibus 1, 2, 3, &c. Rejecto itaque numero dato 54, hinc prodeunt numeratores 1, 2², 3², &c. hoc est 1, 4, 9, &c. Formula etiam Integralis est $\frac{1}{z \times z + 1}$; quare habita ratione numeri 54 quem ex numeratoribus rejecimus, summa Seriei à termino in quo est $z$ in infinitum continuatæ est $\frac{1}{54 \times z + 1}$. Unde summa Seriei integræ est $\frac{1}{108}$. In Serie denique C sunt $n = 7$, $p = 5$, $q = 4$, & primus valor $z = 1$. Unde fit $N = z + 4 \times z + 3 \times z + 2 \times z + 1 - z \times z + 1 \times z + 2 \times z + 3 = 4 \times z + 1$. Valores autem \( N \) per hanc formulam prodeunt semper possunt dividii per \( 4 \times 2 \times 3 \times 4 = 96 \). Ergò hoc divisore rejecto prodeunt numeratores 1, 14, 55, 140, &c. Et formula Summæ, habita ratione numeri 96, est \( \frac{1}{96z} \). Adeoque Summa Seriei integræ est \( \frac{1}{96} \). Scholium 1. Per Propositiones has duas novissimas nullo negotio inveniri possunt Series quot libuerit summabiles. Et vicissim oblatâ Serie hujus speciei, si summari potest, ejus summa plerumque revocatur ad alterutram ex his Propositionibus. In examine tamen solertiâ est opus. Optime autem procedit si termini Seriei oblatæ revocentur ad formulam Prop. III. Sic e.gr. propositâ Serie \( \frac{7}{3 \cdot 5 \cdot 7 \cdot 9 \cdot 11} + \frac{11}{7 \cdot 9 \cdot 11 \cdot 13 \cdot 15} + \frac{15}{11 \cdot 13 \cdot 15 \cdot 17 \cdot 19} + &c. \) Denominatores sic scribi possunt \( 3 \cdot 7 \cdot 11 \times 5 \cdot 9, 7 \cdot 11 \cdot 15 \times 9 \cdot 13, 11 \cdot 15 \cdot 19 \times 13 \cdot 17, &c. \) Unde juxta Prop. III. fit \( n = 4, m = 4, p = 2, q = 1 \), primus valor \( z = 3 \), primus valor \( u = 5 \). Hinc formula Numeratoris invenitur \( 4 \times z + 2u + 8 \), Est autem \( z + 2u + 8 \) semper divisibile per 3; quare rejectis divisoribus datis 4 & 3, per hanc formulam prodeunt Numeratores 7, 11, 15, &c. iidem ac Numeratores in Serie proposta, quæ proinde summabitur per illam propositionem. 2. Cum Series illas \( A, B, C \), communicaveram cum D. Taylor, rescripsit se earum summas inveniile primam quidem \( A \) & tertiam \( C \), eas revocando ad casus simplices Methodi Incrementorum, tertiam \( C \), e.g. revocavit ad hanc formam \( \frac{1}{24} \times \frac{1}{7 \cdot 5} + \frac{1}{5 \cdot 9} + \frac{1}{9 \cdot 13} + \frac{1}{13 \cdot 17} &c. \), ut habeatur summa per praecipia tradita in Scholio Prop. 1. In Serie autem secundâ B, cum hoc non æquè succedit, sequenti usus est Analyti, quam, ipsius venia jam imperata, ob ejus eximiam elegantiam huc transferre non piget. "Seriei istius terminus [in Stylo ejus] cx- "hibetur per formulam \( \frac{z + 2 \times z}{z \times z + 1 \times z \times z + 1} \); pro \( z + 3 \) in denominatore scripto \( z \), quoniam est \( z = 3 \). "Pone \( \frac{B}{C} \) quale esse Integrali quæsito, hoc est \( \frac{B}{C} \) "esse Integrale ipsius \( \frac{z + 2 \times z}{z \cdot z + 1 \times z \cdot z + 1} \), seposito divi- "sore dato 27. Ipsius autem \( \frac{B}{C} \) incrementum est " \( \frac{BC - BC}{CC} \). Debet ergo \( \frac{BC - BC}{CC} \) idem esse ac \( \frac{z + 2 \times z}{z \cdot z + 1 \times z \cdot z + 1} \). Comparando denominatores inveni- "tur C = \( z \times z + 1 \). Hinc itaque sumendo incremen- "ta fit \( C = 2 \times z + z^2 + z \) (= \( 2 \times z + 4 \times z \), quoniam "est \( z = 3 \)). His valoribus in locum \( C & C \) substitu- "tis prodit \( BC - BC = zz + zB - 2 \times z \times z + 2B \), "quod debet esse idem ac \( z + 2 \times z \). Sit \( B = a + v \), "existente \( a \) ipsius \( B \) parte invariabili, & \( v \) parte va- "riabili. Tum sumendo incrementa fit \( B = v \). Unde "ad invenienda \( a \) & \( v \) habetur æquatio \( zz + zv - "2 \times z \times z + 2 \times a + v = z + 2 \times z \), quæ sic scribi "poteat \( z \times z + zv - 2 \times z \times z + 2v = z \times z + 2 \times 1 + 2a \) "vel etiam \( Cv - Cv = z \times z + 2 \times 1 + 2a \). Pone "z + 2a = 0 (unde fit \( a = -\frac{1}{2} \)) & fit \( Cv - Cv = 0 \); "ubi ubi fieri potest \( v = 0 \), (quoniam æquationis termini singuli afficiuntur vel ab \( v \), vel ab \( v \)) Hinc ergò fit \( B = \frac{1}{2} \), adeoque \( \frac{B}{C} = \frac{-1}{2z \times z + 1} \). Unde habitâ ratione divisoris 27, Integrale quæsitum fit \( \frac{-1}{54 \times z \times z + 1} \). Sed & comparando æquationem \( C v - C v = 0 \) cum formulâ generali \( \frac{B C - B C}{CC} = 0 \), inde etiam conclude- re licet esse \( \frac{v}{C} = \text{quantitati datæ}, \) (quoniam ipsius incrementum est 0.) Unde pro \( n \) sumpto quovis numero dato, fit \( v = n C \), atque \( B = \frac{1}{2} + n C \). Quo pacto Integrale quæsitum fit \( \frac{B}{C} = \frac{-\frac{1}{2} + n C}{C} = \frac{-\frac{1}{2}}{C} + n \), quod ab Integrali prius invento differt quantitate datà \( n \). Hoc inde fit, quòd, ut in quadraturâ Curvarum Area inventa augeri potest vel minui area datà, sic in Methodo incrementorum Integrale inven- tum augeri potest vel minui quantitate datà Per Integrale autem primum, ubi deest \( n \), exhibetur summa Scriei in infinitum continuatae. Prop. V. Crescente \( z \) per unitates, & existentibus \( a, b, c, \&c. \) numeris datis integris, quorum nullæ inter se æquantur; invenire Integrale ipsius \( \frac{1}{z \times z + a \times z + b \times z + c \times \&c.} \). Solutio. Ducendo tam numeratorem quam denominatorum fractionis in terminos \( z + 1, z + 2, \&c. \) \( z + a + 1, z + a + 2, \&c. \) \( z + b + 1, z + b + 2, \&c. \) \( z + c + 1, z + c + 2, \&c. \) in denominatore deficien- tes, revocetur Denominator ad formulam \( z \times z + z \). ×z + 2 × &c. denominatoris in Prop. I. Schol. n. 3. Deinde revocetur Numerator ad formam A + Bz + Cz ×z + 1 + Dz × z + 1 × z + 2 + &c. Tum applicando terminos ad Denominatorem novum z × z + 1 ×z + 2 × &c. revocetur fractio ad hanc formam \[ \frac{A}{z \times z + 1 \times &c.} + \frac{B}{z + 1 \times z + 2 \times &c.} + \frac{C}{z + 2 \times z + 3 \times &c.} + \frac{D}{z + 3 \times z + 4 \times &c.} &c. \] Unde denique quaeratur Integrale per Schol. Prop. I. n. 3. Ratio Solutionis per se satis est manifesta. Scholium 1. Hujus Solutionis tota difficultas latet in revocatione numeratoris ad formam requisitam, quod tamen quomodo sit faciendum uno exemplo patebit. Proponatur itaque factum z + 2 × z + 3 × z + 7, quod ad formam propositam sit revocandum. Terminos itaque evolvo gradatim ut sequitur. Factorem primum z + 2 sic scribo 2 + z, cujus terminum primum z duco in 3 + z, unde fit 6 + 2 z: Terminum secundum z duco in 2 + z + 1 (= z + 3) unde fit 2 z + z × z + 1. Dein facia in unam summam colligendo, fit z + 2 × z + 3 = 6 + 2 z + z × z + 1 = 6 + 4 z + z × z + 1. Supereft ut hoc ducatur in z + 7. Itaque terminum primum 6 duco in 7 + z (= z + 7) unde fit 42 + 6 z; terminum secundum 4 z duco in 6 + z + 1 (= z + 7) unde fit 24 z + 4 z × z + 1; terminum tertium z × z + 1 duco in 5 + z + 2 (= z + 7,) unde fit 5 z × z + 1 + z × z + 1 × z + 2. Factis itaque in unum collectis ut prius, fit z + 2 × z + 3 × z + 4 = 42 + 30 z + 9 z × z + 1 + z × z + 1 × z + 2. Et ad eundem modum procedere licet in aliis casibus. 2. Sit 2. Sit autem exemplum Propositionis in fraccióne \[ \frac{1}{z^2 + 2z + 5} \] Restituendo factores \( z + 1, z + 3, z + 4 \) in Denominatore deficientes, fracción fit \[ \frac{z + 1 \times z + 3 \times z + 4}{z^2 + 2z + 3 \times z + 4 \times z + 5} \] Revocandus itaque est Numerator \( z + 1 \times z + 3 \times z + 4 \) ad formam requisitam. Itaque per methodum jam traditam fit primo \( z + 1 \times z + 3 = 1 \times 3 + z + z \times 2 + z + 1 \) \[ = 3 + z + 2z + z \times z + 1 = 3 + 3z + z \times z + 1. \] Deinde \( z + 1 \times z + 3 \times z + 4 = 3 \times 4 + z + 3z \) \[ \times 3 + z + 1 + z \times z + 1 \times 2 + z + 2 = 12 + 3z + 9z \] \[ + 3z \times z + 1 + 2z \times z + 1 + z \times z + 1 \times z + 2 \] \[ = 12 + 12z + 5z \times z + 1 + z \times z + 1 \times z + 2. \] Applicando hoc factum ad Denominatorem \( z \times z + 1 \times \&c. \times z + 5 \) fracción tandem revocatur ad hanc formam \[ \frac{12}{z^2 + 1 \times z + 2 \times z + 3 \times z + 4 \times z + 5} \] \[ + \frac{12}{z + 1 \times z + 2 \times z + 3 \times z + 4 \times z + 5} \] \[ + \frac{5}{z + 2 \times z + 3 \times z + 4 \times z + 5} + \frac{1}{z + 3 \times z + 4 \times z + 5} \] Cujus denique Integrale est \[ \frac{5z \times z + 1 \times z + 2 \times z + 3 \times z + 4}{-12} \] \[ + \frac{4z + 1 \times z + 2 \times z + 3 \times z + 4}{-5} \] \[ + \frac{3z + 2 \times z + 3 \times z + 4}{-1}. \] 3. Quando duo tantum sunt factores \( z & z + 4 \), exhibebitur etiam Integrale per formulam \[ \frac{1}{2} - \frac{1-a}{2z \times z + 1} \] \[ - \frac{1-a \times 2-a}{3z \times z + 1 \times z + 2} - \frac{1-a \times 2-a \times 3-a}{4z \times z + 1 \times z + 2 \times z + 3} \&c. \] Seriem nempe continuando donec abrupmtatur per evanescentiam nescentiam terminorum. Si Factores duo sint \( z \) & \( z - a \) exhibebitur Integrale per formulam \( \frac{1}{z-1} - \frac{1+a}{2 \cdot (z-1)(z-2)} \) \( - \frac{1+a}{3 \cdot (z-1)(z-2)(z-3)} + \ldots \). Potest idem Integrale exprimi utroque modo, prout fractionis oblatæ factor vel minor vel major sumatur pro \( z \). 4. Si primus valor \( z \) sit \( a + 1 \), migrabit formula posterior in hanc \( \frac{1}{a} \times \frac{1}{1} \times \frac{1}{2} \times \frac{1}{3} + \ldots \) usque \( \frac{1}{a} \) inclusivè, quâ, cum signo contrario, exhibetur summa Seriei \( \frac{1}{1 \times 1 + a} + \frac{1}{2 \times 2 + a} + \frac{1}{3 \times 3 + a} + \ldots \) in infini- tum continuatæ. Sit e. gr. \( a = 1 \), atque Series erit \( \frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + \ldots = \frac{1}{1} \times \frac{1}{1} = 1 \). Si \( a = 2 \), erit Series \( \frac{1}{1 \times 3} + \frac{1}{2 \times 4} + \frac{1}{3 \times 5} + \ldots = \frac{1}{2} \times \frac{1}{1} + \frac{1}{2} = \frac{3}{4} \); Si \( a = 3 \), Series erit \( \frac{1}{1 \times 4} + \frac{1}{2 \times 5} + \frac{1}{3 \times 6} + \frac{1}{4 \times 7} + \ldots \) \( = \frac{1}{3} \times \frac{1}{1} + \frac{1}{2} + \frac{1}{3} = \frac{11}{18} \). 5. Ex eâdem Serie \( \frac{1}{1 \times 1 + a} + \frac{1}{2 \times 2 + a} + \frac{1}{3 \times 3 + a} + \ldots \) \( + \ldots \) pro diverso valore \( a \) oriuntur Series plures formâ satis elegantès, quarum nonnullas Lectori ob oculos sistere, credo, ingratum non erit. Si pro \( a \) sumantur succedivè numeri pares, 2, 4, 6, 8, \&c. Series erunt Si \( a = 2 \): \[ \frac{1}{1 \times 1 + 2} + \frac{1}{2 \times 2 + 2} + \frac{1}{3 \times 3 + 2} + \frac{1}{4 \times 4 + 2} + \ldots \] 4) \( \frac{1}{1 \times 1 + 4} + \frac{1}{2 \times 2 + 4} + \frac{1}{3 \times 3 + 4} + \frac{1}{4 \times 4 + 4} + \ldots \) 5) \( \frac{1}{1 \times 1 + 6} + \frac{1}{2 \times 2 + 6} + \frac{1}{3 \times 3 + 6} + \frac{1}{4 \times 4 + 6} + \ldots \) 5) \( \frac{1}{1 \times 1 + 8} + \frac{1}{2 \times 2 + 8} + \frac{1}{3 \times 3 + 8} + \frac{1}{4 \times 4 + 8} + \ldots \) Vel Vel $\frac{1}{4-1} + \frac{1}{9-1} + \frac{1}{16-1} + \frac{1}{25-1} + \&c.$ $\frac{1}{9-4} + \frac{1}{16-1} + \frac{1}{25-4} + \frac{1}{36-4} + \&c.$ $\frac{1}{16-9} + \frac{1}{25-9} + \frac{1}{36-9} + \frac{1}{49-9} + \&c.$ $\frac{1}{25-16} + \frac{1}{36-16} + \frac{1}{49-16} + \frac{1}{64-16} + \&c.$ Vel $\frac{1}{4-1} + \frac{1}{9-1} + \frac{1}{16-1} + \frac{1}{25-1} + \&c.$ $\frac{1}{4+1} + \frac{1}{9+3} + \frac{1}{16+5} + \frac{1}{25+7} + \&c.$ $\frac{1}{4+3} + \frac{1}{9+7} + \frac{1}{16+11} + \frac{1}{25+15} + \&c.$ $\frac{1}{4+5} + \frac{1}{9+11} + \frac{1}{16+17} + \frac{1}{25+23} + \&c.$ Si pro $a$ sumatur successive numeri impares $1, 3, 5, 7, \&c.$ Series erunt 1) $\frac{1}{1 \times 1 + 1} + \frac{1}{2 \times 2 + 1} + \frac{1}{3 \times 3 + 1} + \frac{1}{4 \times 4 + 1} + \&c.$ 3) $\frac{1}{1 \times 1 + 3} + \frac{1}{2 \times 2 + 3} + \frac{1}{3 \times 3 + 3} + \frac{1}{4 \times 4 + 3} + \&c.$ 5) $\frac{1}{1 \times 1 + 5} + \frac{1}{2 \times 2 + 5} + \frac{1}{3 \times 3 + 5} + \frac{1}{4 \times 4 + 5} + \&c.$ 7) $\frac{1}{1 \times 1 + 7} + \frac{1}{2 \times 2 + 7} + \frac{1}{3 \times 3 + 7} + \frac{1}{4 \times 4 + 7} + \&c.$ Vel $\frac{1}{2} \times \frac{1}{1} + \frac{1}{3} + \frac{1}{6} + \frac{1}{10} + \&c.$ $\frac{1}{2} \times \frac{1}{3-1} + \frac{1}{6-1} + \frac{1}{10-1} + \frac{1}{15-1} + \&c.$ $\frac{1}{2} \times \frac{1}{6-3} + \frac{1}{10-3} + \frac{1}{15-3} + \frac{1}{21-3} + \&c.$ $\frac{1}{2} \times \frac{1}{10-6} + \frac{1}{15-6} + \frac{1}{21-6} + \frac{1}{28-6} + \&c.$ Vel $\frac{1}{2} \times \frac{1}{1+0} + \frac{1}{3+0} + \frac{1}{6+0} + \frac{1}{10+0} + \&c.$ $\frac{1}{2} \times \frac{1}{1+1} + \frac{1}{3+2} + \frac{1}{6+3} + \frac{1}{10+4} + \&c.$ $\frac{1}{2} \times \frac{1}{1+2} + \frac{1}{3+4} + \frac{1}{6+6} + \frac{1}{10+8} + \&c.$ $\frac{1}{3} \times \frac{1}{1+3} + \frac{1}{3+6} + \frac{1}{6+9} + \frac{1}{10+12} + \&c.$ 6. Ante aliquot annos D. Jac. Bernoulli Geometra insignis invenit summam Seriei cujuslibet, cujus Numeratores constituunt Seriem æqualium, Denominatores vero constituunt, vel Seriem quadratorum dato aliquo quadrato \( Q \) minutorum, vel Seriem Triangulorum, dato aliquo Triangulo \( T \) minutorum. Hæc invenit ille observando quod hujusmodi Series oriuntur ex ablatione Seriei Harmonicè proportionalium truncatæ ab eadem Serie integra; nempe ita ut numerus terminorum deficientium in Serie truncata, sit, vel duplus lateris dati quadrati \( Q \), vel duplus unitate auctus lateris dati Trianguli \( T \). Idem etiam observavit frustra quaeri summam Seriei reciprocae Quadratorum. Hoc idem etiam verum est de reciprocis Cuborum, vel aliarum quarumlibet dignitatum numerorum in progressionе Arithmetica. Ratio est, quòd nulla intercedit differentia inter factores denominatorum, quod ad hujusmodi summationes semper requiri constat ex Methodo sumendi differentias in Scholio Prop. I. jam explicatæ. Nam si per formulam aliquam exhiberi possit summa quaesita, differentia istius formulae exhiberet terminos Seriei propositæ: sed in tali differentiâ denominator semper afficitur per factores ab invicem diversos, quod quo- niam in Seriebus prædictis non obtinet, summæ Serierum hujusmodi in terminis finitis haberi nequeunt. Ad eundem fere modum, argumento petito à Prop. III. & IV. demonstrari potest summæ Serierum exhiberi non posse in terminis numero finitis, quarum Numeratores constituunt Seriem æqualium. Denominatores vero constant ex certo numero terminorum in progressionе Arithmetica, maximo factore cujusvis termini minore existente quàm factor minimus in termino proxime sequenti, cujusmodi est Series \( \frac{1}{1 \cdot 2} + \frac{1}{3 \cdot 4} + \frac{1}{5 \cdot 6} + \cdots \). 7. Jam liceret regulas nonnullas tradere quas pro casibus quibusdam singularibus concinnavi; sed hæc nos nos longius abducerent. Sufficiat itaque quae generalliora sunt explicatis, & simul monstrate, ad novae hujus Serierum infinitarum doctrinae provectionem nihil magis facere, quam si excogitentur formule generaliores summarum, ex quarum differentiis, per regulas supra traditas computatis, deinde conficiantur Canones quantitatum summabilium; ita ferè ut jam factum est in Calculo Integrali, b. e. in Stylo Newtoniano, in Methodo Fluxionum. 8. Restituendo factores in Denominatore deficientes potuisset praesens Problema revocari ad Propositionem II. Sed & in terminis generalioribus proponi potest, nempe pro Numeratore sumptâ quâvis Formulâ, cujus differentia aliqua datur. Sub ea tamen conditione ut dimensiones Denominatoris ad minimum binario superent Dimensiones Numeratoris: aliàs enim summa Seriei in terminis numero finitis haberi nequit. Sit hujus rei exemplum in Serie $\frac{1}{1 \cdot 3 \cdot 5 \cdot 7} + \frac{4}{2 \cdot 4 \cdot 6 \cdot 8}$ $\frac{9}{3 \cdot 5 \cdot 7 \cdot 9} + \frac{16}{4 \cdot 6 \cdot 8 \cdot 10} + \&c.$ ubi Numeratores sunt numerorum naturalium quadrata. Applicando tum Numeratores tum Denominatores ad numeros naturales, Series revocatur ad formam simpliciorem $\frac{1}{3 \cdot 5 \cdot 7} + \frac{2}{4 \cdot 6 \cdot 8}$ $\frac{3}{5 \cdot 7 \cdot 9} + \frac{4}{6 \cdot 8 \cdot 10} + \&c.$ Per $p$ designatis numeris naturalibus 1, 2, 3, 4, &c. terminus Seriei designatur per formulam $\frac{p}{p + 2 \times p + 4 \times p + 6}$; vel per formulam $\frac{z - 2}{z \times z + 2 \times z + 4}$, nempe pro $p + 2$ scripto $z$. Quamquam progrediendo de termino in terminum augetur $z$ per unitates, restituendi sunt factores in denominatore deficientes $z + 1$, $z + 3$, & hoc pacto revocatur terminus Seriei ad formulam $\frac{z - 2 \times z + 1 \times z + 3}{z \times z + 1 \times z + 2 \times z + 3 \times z + 4}$ Per methodum in hâc Propositione jam explicatam vocatur M m m m m vocatur numerator ad formam $-6 - 6z - z \times z + 1 + z \times z + 1 \times z + 2$. Unde habita ratione denominatoris Terminus revocatur ad formam $\frac{-6}{z \times z + 1 \times z + 3 \times z + 4}$ $+ \frac{1}{z + 1 \times z + 2 \times z + 3 \times z + 4} + \frac{1}{z + 2 \times z + 3 \times z + 4}$ $+ \frac{1}{z + 3 \times z + 4}$. Adeoque sumendo Integrale fit $\frac{5}{4z \times z + 1 \times z + 2 \times z + 3} + \frac{1}{3 \times z + 1 \times z + 2 \times z + 3}$ $+ \frac{1}{2 \times z + 2 \times z + 3} + \frac{1}{z + 3}$; quo, sub signo contrario, exhibetur summa Seriei in infinitum continuatae, incipientis à termino $\frac{\xi - 2}{\xi \times \xi + 2 \times \xi + 4}$. Summa itaque Seriei integrae incipientis à termino $\frac{1}{3 \cdot 5 \cdot 7}$ est $\frac{31}{240}$. Si per Prop. II. procedere esset animus, ex formulâ $z - 2 \times z + 1 \times z + 3$ collectis numeratoribus primis 24, 70, 144, 252, sumendo sorum differentias habeantur $46 = b$, $28 = c$, $6 = d$, $e = o = \&c.$ existente $M = 24$: unde per Lem. 2. prodiret formula $-6 - 6z - z \times z + 1 + z \times z + 1 \times z + 2$, quâ designatur Terminus, eadem ac supra; atque pergendo per Prop. II. haberetur summa. Prop. VI. Prob. Invenire summam quotlibet terminorum Seriei Fractionum, quarum Numeratores & Denominatores constituunt lineas duas qualvis transversas in Triangulo Arithmetico Paschaltii; nempe cujus generatores sunt unitates. Solutio. Per $n$ designetur Ordo Seriei Numeratorum in Triangulo Arithmetico, & sit $p$ differentia inter ordinem Numeratorum & Denominatorum, & per $q$ designetur numerus terminorum quorum summa requiritur quiritur. Tum si Denominatores sint plurium dimensionum quam sunt Numeratores, Summa exhibebitur per formulam primam sequentem; si dimensiones Numeratorum plures sunt quam dimensiones Denominatorum, Summa exhibebitur per formulam secundam. **Formula I.** \[ \frac{n + p - 1}{p - 1} = \frac{n \cdot n + 1 \cdot n + 2 \cdot \&c. \cdot n + p - 1}{p - 1 \times n + q \cdot n + q + 1 \cdot \&c. \cdot n + q + p - 2} \] **Formula II.** \[ \frac{n - p - 1}{p + 1} = \frac{q + n - 1 \cdot q + n - 2 \cdot \&c. \cdot q + n - p - 1}{p + 1 \times n - 1 \cdot n - 2 \cdot \&c. \cdot n - p} \] **Ex. 1.** Inveniendum sit aggregatum sex primorum terminorum Seriei \( \frac{1}{1} + \frac{4}{7} + \frac{10}{28} + \frac{20}{84} + \frac{35}{210} + \frac{56}{462} + \&c. \) ubi Numeratores constituunt lineam quartam, Denominatores constituunt lineam septimam in Triangulo Arithmetico Sunt itaque \( n = 4, p = 3, q = 6 \); & quoniam dimensiones Denominatorum superant dimensiones Numeratorum, dabitur summa per Formulam primam; nempe \( \frac{4 + 3 - 1}{3 - 1} = \frac{4 \cdot 5 \cdot 6}{3 - 1 \times 4 + 6 \times 4 + 7} \) sive \( 3 - \frac{6}{11} = 2 \frac{5}{11} \). **Ex. 2.** Quæratur summa sex primorum terminorum Seriei \( \frac{1}{1} + \frac{7}{4} + \frac{28}{10} + \frac{84}{20} + \frac{210}{35} + \frac{462}{56} + \&c. \) cujus termini sunt terminorum Seriei prioris reciproci. Sunt itaque \( n = 7, p = 3, q = 6 \), adeoque per formulam secundam summa fit \( \frac{3}{4} + \frac{12 \cdot 11 \cdot 10 \cdot 9}{4 \times 6 \cdot 5 \cdot 4} = 24 \). **Scholium 1.** Formulas in hac propositione exhibitas ante biennium communicavi cum Viris celeberrimis Moivre & Bernoullis. Facile autem derivari possunt ex præceptis in Prop. I. traditis. Sit exemplum in Serie priori \( \frac{1}{1} + \frac{4}{7} + \frac{10}{28} + \&c. \). Per \( p \) designato loco Ter- Termini in Serie hâc, exhibetur Terminus per formulam \[ \frac{4}{p + 3} + \frac{5}{p + 4} + \frac{6}{p + 5} \] Unde regrediendo ad Integrale, summa Series incipientis à termino illo exhibetur per formulam \[ \frac{4 \cdot 5 \cdot 6}{2 \cdot p + 3 \cdot p + 4} \] adeoque pro \( p \) sumpto 1, Series integra fit \[ \frac{4 \cdot 5 \cdot 6}{2 \cdot 4 \cdot 5} = 3, \] atque summa primorum sex terminorum fit \[ \frac{4 \cdot 5 \cdot 6}{2 \cdot 10 \cdot 11}, \] omnino ut per formulam jam exhibetur. 2. In formulâ primâ summa Series in infinitum continuatæ est \[ \frac{n + p - 1}{p - 1}, \] evanescente jam parte alterâ formulæ. Sed in casu formulæ secundæ summa hæc est infinitum quid, cujus species, respectu numeri infiniti \( q \), exhibetur per formulæ partem alteram, quæ in hoc casu fit \[ \frac{q + 1}{p + 1 \times n - 1 \cdot n - 2 \cdot \&c. \cdot n - p}. \] 3. De hujusmodi Seriebus in epistolâ datâ mensâ Maio 1716, sic ad me scripsit Vir. Ill. D. Leibnitius, quem magno Scientiarum damno nobis nuper erectum lugemus. "Il me semble qu’autrefois j’ay aussî sommé quelques Series ou suites comme \[ \frac{1}{1} + \frac{2}{4} + \frac{3}{10} + \frac{4}{20} \] \[ + \frac{5}{35} + \frac{6}{56} + \&c. \] Le terme de cette suite exprimé analytiquement est \[ x \cdot x + 1 \cdot x + 2 \cdot \frac{1}{1} \cdot \frac{1}{2} \cdot \frac{1}{3} \] \[ = \frac{1 \cdot 2 \cdot 3}{x + 1 \cdot x + 2} = \frac{6}{xx + 3x + 2}. \] On demande donc la somme d’une suite donnée, dont un terme soit \[ \frac{1}{xx + 3x + 2} \] ou \( x \) signifie les nombres naturales \[ 1, 2, 3, 4, \&c. \] & l signifie l’Unité, ou la difference des \( x \). Supposons que le terme de la suite som- matrice matrice demandée soit \( \frac{fx}{mx + n} = \frac{\circ}{D} \). Or Diff. \( \frac{\circ}{D} = \) \[ \frac{\circ}{D} + \frac{\circ + d\circ}{D + dD} = \frac{d\circ - \circ dD}{D D + D dD} : \text{ sed } d\circ = f dx, \] \& \( dD = m dx = ml; \) donc la Difference de \( \frac{\circ}{D} \) est \( = \) \[ \frac{nfl}{mxx + 2mnix + nnil} + \frac{mfl}{mxx + 3mnlx + 2mnl} \] Maintenant il faut faire \[ \frac{nfl}{mxx + 2mnix + nnil} = \frac{mfl}{mxx + 3mnlx + 2mnl} \] c'est à dire, il faut identifier ces deux formules, ou la donnée est Multipliée par \( \frac{nf}{mm} \): donc égalant les termes respectifs, puisque les \( x \) conviennent, on aura par les \( x \), \( 2n + m = 3m \), c'est adire il y aura \( m = n \), & par les absolus on aura \( n n + mn = 2m m \), ce qui donne encore \( m = n \); donc l'identification réussit, & nous pouvons faire \( n = m = l = 1 \), & \( f = 1 \) (car \( f \) demeure arbitraire) & le terme de la suite sommatrice sera \( \frac{x}{x + 1} \), car diff. \( \frac{x}{x + 1} \) donne \[ -\frac{x}{x + 1} + \frac{x + 1}{x + 2} = \frac{1}{xx + 2x + 2}, \text{ & par conséquent } \] \[ \frac{6x}{x + 1} \text{ donne la somme des } \frac{x}{xx + 1x + 2x + 2}. \] \( 3, 4, \frac{9}{2}, \frac{24}{5}, 5, \frac{36}{7}, \text{ &c. Series summatrix, cujes sero- } \) minus \( \frac{6x}{x + 1} \). \[ \frac{1}{1} + \frac{2}{4} + \frac{3}{10} + \frac{4}{20} + \frac{5}{35} + \text{ &c. Series summanda, cu- } \] jus terminus \( \frac{x}{xx + 1x + 2x + 2x + 2} \). Et pour s'en servir aux formations, les 5 termes, par Ex. de N n n n n la suite donnée seront $\frac{36}{7} - 3 = \frac{15}{7}$. Et généralement la somme des termes jusqu'à quelque terme $x$ exclusivement, sera $\frac{6x}{x + 1}$. $- 3$: Et pour la somme de la suite entière à l'infinie, $x$ devient infini, & $\frac{6x}{x + 1} = 6$: donc la somme de toute la suite est $6 - 3 = 3$, comme vous l'avez trouvé. Cette méthode est le calcul des différences appliqué aux Nombres; & il faut vous avouer qu'avant que de l'appliquer aux Figures, & même avant que d'avoir été Géomètre, Je le pratiquai en quelque façon dans les nombres; ayant trouvé encore jeune garçon que les suites dont les Numerateurs fussent des Unites, & dont les Denominateurs fussent les Nombres figurés, comme Triangulaires Pyramidaux &c. étoient les différences 1ères, 2èmes, 3èmes, &c. multipliées par les constantes de la suite $\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + &c.$ & par conséquent sommables. Mais quand je devins un peu Géomètre & Analyste, Je vis qu'il y avait moyen de venir à bout de telles formations par une Methode générale, autant qu'il étoit possible; & que le calcul des différences étoit encore plus commode dans la Géométrie que dans les Nombres, puis qu'il y a plus d'évanouissements, & que les différences répondent aux Tangentes, les sommes aux Quadratures. Cette méthode générale de chercher la suite sommatrice de la suite donnée, quand elle est possible, résulte toujours, quand le terme de la suite donnée exprimé Arithmétiquement n'a point la quantité variable enveloppée dans une racine, ny entrant dans l'exposant; & alors, on peut tou- jours jours determiner la suite sommatrice, ou prouver qu'il est impossible d'en trouver. Et la chose réussit même bien souvent, lors même que la variable entre dans l'Exposant. Mais comme il y a quelques fois des Quadratures particulières de quelques portions d'une Figure, dont ou ne sauroit donner la Quadrature generelle ou la Figure quadratrice ; de même on peut trouver quelquefois la somme de toute la suite, ou d'un certaine partie, quoys qu'on ne puisse pas trouver la somme de chaque partie ; & alors il faut avoir recours à des Methodes particulières, dont on n'est pas toujours le maître, nostre Analyse n'étant pas encore portée à sa perfection. Prop. VII. Prob. Invenire summam Seriei cujus Numeratores consti- tuunt lineam quamlibet erectam in Triangulo Arith- metico Paschali, Denominatores vero constituunt li- neam quamlibet transversam. Solutio. Designetur ordo lineæ erectæ per \( p \), ordo lineæ transversæ per \( q \), & sit \( m \) aggregatum tot terminorum primorum in lineâ erectâ ordinis \( p + q - 1 \) quot sunt unitates in \( q - 1 \), atque summa quaesita erit \[ \frac{2^{p+q-2} - m}{p \cdot p + 1} \times \frac{1 \cdot 2 \cdot 3 \cdot \ldots \cdot (q - 1)}{p \cdot p + q - 2}. \] Ex. 1. Proponatur Series \( \frac{1}{1} + \frac{5}{4} + \frac{10}{10} + \frac{10}{20} + \frac{5}{35} + \frac{1}{56} \) Ubi Numeratores constituunt lineam sextam erectam, Denominatores occupant lineam quartam transversam. In hoc itaque casu sunt \( p = 6, q = 4, p + q - 1 = 9, q - 1 = 3 \), adeoque \( m = 1 + 8 + 28 = 37 \) i.e. tribus terminis primis lineæ nonæ erectæ. Unde fit summa quaesita \( \frac{2^8 - 37}{6 \cdot 7 \cdot 8} = \frac{219}{56} \). Ex. 2. Constituant Numeratores lineam centesimam erectam, & sint Denominatores Numeri Trigonales, qui occupant lineam tertiam transversam. Tum erunt \( p = 100, q = 3, m = 102 \) atque adeo summa quaestà fit \[ \frac{2^{101} - 102}{100 \cdot 101} \times \frac{1 \cdot 2}{100 \cdot 101}. \] Cor. Si \( q = 2 \), formula fit \( \frac{2^p - 1}{p} \), quâ exhibetur ag- gregatum primi termini, unâ cum semisè secundi, triente tertii, quadrante quarti, & sic porrò, linea cu- jusvis erectae ordinis \( p \) Trianguli Arithmetici Paschalli. Sic v.gr. est \( \frac{1}{1} + \frac{5}{2} + \frac{10}{3} + \frac{10}{4} + \frac{5}{5} + \frac{1}{6} = \frac{2^6 - 1}{6} = 10 \frac{1}{2} \). Prop. VIII. Prob. Invenire summam ejusdem Seriei, quando terminorum signa sunt alternatim \( + \) & \( - \). Solutio. Summa quaestà exhibetur per formulam sim- plicissimam \( \frac{q - 1}{p + q - 2} \). Ex. Invenienda sit summa Seriei \( \frac{1}{1} - \frac{6}{9} + \frac{15}{45} - \frac{20}{165} \) + \( \frac{15}{495} - \frac{6}{1287} + \frac{1}{3003} \), ubi Numeratores constituunt li- neam septimam erectam, Denominatores constituunt nonam transversam. In formulà itaque pro \( p \) & \( q \) scrip- tis 7 & 9, sit summa \( \frac{8}{14} \). Manente eadem Serie Numeratorum (sempe lineâ sep- timâ erectâ), si pro Serie Denominatorum iumentur successivè lineæ transversæ 2\(^{st}\), 3\(^{rd}\), 4\(^{th}\), &c. Summae erunt \( \frac{1}{7}, \frac{2}{8}, \frac{3}{9}, \frac{4}{10}, \frac{5}{11}, \&c. \) quæ sic possent scribi, \( \frac{1}{7}, \frac{7}{28}, \frac{28}{84}, \frac{84}{210}, \frac{210}{462}, \&c. \) ubi tam Numeratores, quam Denominatores excerptur ex lineâ transversâ ordinis septimi. Idem eveniret si loco septimi Numeratores constituiscent aliam quamlibet lineam erectam ordinis \( p \); Summae quippe oriuntur ex applicatione terminorum lineæ lineae transversae ejusdem ordinis \( p \) ad terminos proximè sequentes in eadem linea. Propositiones hæ duæ novissimæ potius elegantæ sunt quàm utiles; quare Formularum nostrarum demonstrationem Lectoris solertia investigandam relinquimus, ad Propositionem ultimam jam properantes, quæ tertiam continet Serierum speciem, ob usum multiplicem fatis insignem. Lemma 5. Sit Series quævis \( \frac{M}{b}, \frac{N}{b^2}, \frac{O}{b^3}, \frac{P}{b^4}, \&c. \) cujus terminorum Denominatores constituunt progressionem quamlibet Geometricam \( b, b^2, b^3, b^4, \&c. \) Sint etiam Numeratorum primus \( A (= M) \), prima differentiarum primarum \( B \), prima secundarum \( C \), prima tertiarium \( D \), quartarum \( E \), & sic porrò; & sint \( \frac{\alpha}{b}, \frac{\beta}{b^2}, \frac{\gamma}{b^3}, \frac{\delta}{b^4}, \&c. \) respectivè, aggregata, Unius, Duorum, Trium, Quatuor, vel plurium terminorum Seriei \( \frac{M}{b}, \frac{N}{b^2}, \frac{O}{b^3}, \&c. \) atque sint Numeratorum primus \( a (= \alpha) \) prima differentiarum primarum \( b \), prima secundarum \( c \), prima tertiarium \( d \), & sic porrò: & sit \( b - 1 = q \). Tum ipsorum \( a, b, c, d, \&c. \) valores erunt. \[ \begin{align*} a &= A = \alpha = M \\ b &= bA + B \\ c &= q bA + bB + C \\ d &= q^2 bA + q bB + bC + D \\ &\text{& sic porrò.} \end{align*} \] Demonstratio. Satis constat esse \( a = \alpha = A = M \). Termini \( \frac{M}{b}, \frac{N}{b^2}, \frac{O}{b^3}, \frac{P}{b^4}, \&c. \) Numeratoribus \( M, N, O, P, \&c. \) &c. expressis per $A$, $B$, $C$, $D$, &c. transformantur in terminos $\frac{A}{b}$, $\frac{A + B}{b^2}$, $\frac{A + 2B + C}{b^3}$, $\frac{A + 3B + 3C + D}{b^4}$ &c. Unde colligendo summas terminorum, inveniuntur Numeratores $\alpha$, $\beta$, $\gamma$, $\delta$, &c. nempe $\alpha = \frac{A}{b}$ $\beta = \frac{A + B}{b^2}$ $\gamma = \frac{A + 2B + C}{b^3}$ $\delta = \frac{A + 3B + 3C + D}{b^4}$ &c. Unde sumendo differentias fiunt $b = q b A + h B$ $c = q b A + h B + h C$ $d = q q b A + q b B + h C + D$ & sic porro, ut in Propositione exhibentur. Cor. 1. Si Numeratorum $M$, $N$, $O$, $P$, &c. differentia vel prima, vel secunda, vel alia quaedam detur, terminis omnibus post primos aliquot in Serie $A$, $B$, $C$, $D$, &c. evanescentibus, Differentiae $b$, $c$, $d$, &c. tandem incurrent in Progressionem Geometricam in ratione $r$ ad $q$. Exempli gratia, si detur Numeratorum $M$, $N$, $O$, $P$ &c. differentia prima $B$, erunt $c$, $d$, &c. in ratione continuâ Geometricâ $r$ ad $q$; ut constat per illorum valores $q b A + h B$, $q q b A + q b B$, &c. existentibus $C = o = D = &c.$ Cor. 2. Ordo autem primæ differentiarum $B$, $C$, $D$, &c. quae hoc modo evanescunt, idem est ac ordo differentiae vel $b$, vel $c$, &c. unde incipit Progressio illa Geometrica. Sic si $B = o = C = &c.$ erunt $b$, $c$, $d$, &c. in Progressione Geometricâ; si $C = o = D = &c.$ erunt $c$, $d$, &c. in Progressione Geometricâ. Et sic porro. Lemma 6. Iisdem positis fit $r$ terminus unde incipit Progressio Geometrica in Serie differentiarum $b$, $c$, $d$, &c. & per $p + r$ \( p + 1 \) designetur ordo Termini in Serie \( \frac{a}{b}, \frac{\beta}{b^2}, \frac{\gamma}{b^3}, \frac{\delta}{b^4} \), &c. Tum Terminus ille designabitur per fractionem cujus Denominatore existente \( b^{p+1} \) Numerator est \[ \frac{a + bp + cp \times \frac{p-1}{2} + dp \times \frac{p-1}{2} \times \frac{p-2}{3} + \&c. + \frac{p}{q}}{b^p - 1 - qp - q^2 p \times \frac{p-1}{2} - q^3 p \times \frac{p-1}{2} \times \frac{p-2}{3} - \&c.} \] nempe per \( n \) designato ordine differentiae evanescentis in Serie B, C, D, &c. ut & Numero terminorum \( a + bp, \&c. \) item terminorum \( -1 - qp, \&c. \). Demonstratio. Per Lemma 1. Termini istius Numerator exhibetur per formulam \[ a + bp + cp \times \frac{p-1}{2} + dp \times \frac{p-1}{2} \times \frac{p-2}{3} + \&c. (p + x \text{ subeunte vices } x \text{ in Lemmate isto}) \] Ergò si sit, ex gr. \( n = 2 \), per Lemm. 5. Cor. 2. erunt \( c, d, \&c. \) in ratione continuâ \( r \) ad \( q \). Numerator itaque in hoc casu est \[ a + bp + cp \times \frac{p-1}{2} + cq p \times \frac{p-1}{2} \times \frac{p-2}{3} + cq^2 p \times \frac{p-1}{2} \times \frac{p-2}{3} \times \frac{p-3}{4} + \&c. \text{ Sed si termini } cp \times \frac{p-1}{2} + cq p \times \frac{p-1}{2} \times \frac{p-2}{3} + \&c. \text{ ducantur in } \frac{q^2}{c}, \&c. \text{ producuntur termini } 1 + qp, \text{ prodibit Series quà exprimitur binomii } 1 + q \text{ dignitas } 1 + q^p = b^p. \text{ Ergo productum illud æquale est } b^p - 1 - qp; \text{ adeoque termini } cp \times \frac{p-1}{2} + cq p \times \frac{p-1}{2} \times \frac{p-2}{3} + \&c. = \frac{c}{q^2} \times b^p - 1 - qp. \text{ Quo pacto Numerator fit } a + bp + \frac{c}{q^2} \times b^p - 1 - qp, \text{ existentibus duobus terminis } a + lp, \text{ ut & duobus } -1 - qp, \text{ juxta sensum Propositionis, quoniam } n = 2. \text{ Atque cadem est demonstratio in aliis casibus. De Denominatore verò per se satis conilit.} Prop. Prop. IX. Prob. Invenire summam quotlibet terminorum Series cujusvis \( \frac{M}{h}, \frac{N}{h^2}, \frac{O}{h^3}, \frac{P}{h^4}, \ldots \) cujus terminorum Denominatores constituunt progressionem quamlibet Geometricam \( h, h^2, h^3, h^4, \ldots \). Numeratores autem sunt quantitates differentiâ aliquà conitanti gaudentes. Solutio Sunto Numeratorum \( M, N, O, P, \ldots \) primus \( A \), prima differentiarum primarum \( B \), prima secundarum \( C \), prima tertiarium \( D \), & sic porrò; & sit ipsorum \( A, B, C, D, \ldots \) numerus \( n \), atque \( h - 1 = q \). Tum fiat \( a = A (= M) b = hA + B, c = qhA + hB + C, d = q^2hA + qhB + hC + D, \ldots \) ut sint tot termini \( a, b, c, d, \ldots \), quot sunt unitares in \( n + 1 \). Terminorum istorum ultimus dicatur \( r \), atque per \( p + 1 \) designetur numerus terminorum \( \frac{M}{h}, \frac{N}{h^2}, \frac{O}{h^3}, \frac{P}{h^4}, \ldots \) quorum summa requiritur; Dico summam illam exhiberi per fractionem, cujus Denominatore existente \( h^{p+1} \), Numerator est \[ a + bp + cp \times \frac{p-1}{2} + dp \times \frac{p-1}{2} \times \frac{p-2}{3} + \ldots + \frac{r}{q^n} \times h^p - 1 - qp - q^2p \times \frac{p-1}{2} - q^3p \times \frac{p-1}{2} \times \frac{p-2}{3} - \ldots - \frac{r}{q^n} \] Demonstratio. Nam (per Lem. 6.) per hanc formulam repræsentatur terminus ordine \( p + 1 \) Seriei \( \frac{\alpha}{h}, \frac{\beta}{h^2}, \frac{\gamma}{h^3}, \frac{\delta}{h^4}, \ldots \) qui terminus (per constructionem Lemmatis 5.) xqualis est aggregato terminorum numero \( p + 1 \) Seriei propositæ \( \frac{M}{h}, \frac{N}{h^2}, \frac{O}{h^3}, \frac{P}{h^4} \). Q.E.D. Ex. I. Ex. 1. Invenienda sit summa novem terminorum Seriei \( \frac{1}{2}, \frac{2}{4}, \frac{3}{8}, \frac{4}{16}, \ldots \). Sunt in hoc casu \( b = 2 \), \( q = b - 1 \) \( = 1 \), \( p + 1 = 9 \), \( p = 8 \), \( A = 1 \), \( B = 1 \), \( C = 0 \), \( D = \ldots \), adeoque \( n = 2 \), (quoniam sunt duo \( A, B, \)) Hinc fit \( a = A = 1 \), \( b = hA + B = 2 \times 1 + 1 = 3 \), \( c = qhA + hB + C = 2 \times 1 + 2 \times 1 + 0 = 4 = r \), Adeoque per formulam fit summa quaestà \[ \frac{1 + 3 \times 8 + \frac{4}{1} \times 2^3 - 1 - 1 \times 8}{2^9} = \frac{1013}{512}. \] Ex. 2. Quæratur summa sex terminorum Seriei \( 1 \times 3 + 3 \times 3^2 + 6 \times 3^3 + 10 \times 3^4 + 15 \times 3^5 + 21 \times 3^6 + \ldots \). In hoc casu sunt \( b = \frac{1}{3} \), \( q = \frac{-2}{3} \), \( p + 1 = 6 \), \( p = 5 \), \( A = 1 \), \( B = 2 \), \( C = 1 \), \( D = 0 = E = \ldots \), adeoque \( n = 3 \), atque \( a = 1 \), \( b = \frac{1}{3} + 2 = \frac{7}{3} \), \( c = \frac{-2}{9} + \frac{2}{3} + 1 = \frac{13}{9} \), \( d = \frac{4}{27} - \frac{4}{9} + \frac{1}{3} = \frac{1}{27} = r \). Unde summa quaestà fit \( = 19956 \). sive \[ \frac{1 + \frac{7}{3} \times 5 + \frac{13}{9} \times 5 \times \frac{4}{2} + \frac{-1}{8} \times \frac{1}{3}^5 - 1 + \frac{2}{3} \times 5 - \frac{4}{9} \times 5 \times \frac{4}{2}}{\frac{1}{3}}. \] Cor. 1. Ejusdem Seriei, à termino primo \( \frac{M}{b} \) in infinitum continuatæ, summa exhibetur per formulam simplicissimam \( \frac{A}{b - 1} + \frac{B}{(b - 1)^2} + \frac{C}{(b - 1)^3} + \frac{D}{(b - 1)^4} + \ldots \). Cor. 2. Si \( b = 2 \), Seriei totius in infinitum continuatæ summa habetur solâ additione terminorum \( A, B, C, D, \ldots \). Et hæc summa eadem est ac summa lineæ erectæ respondentis termino primo \( A \), in Triangulo Arithmetico, cujus lineam transversam occupant Numeratores ratores \( M, N, O, P, \&c. \) Quod facile constat ex contemplatione Trianguli. Si itaque fuerint \( M, N, O, \&c. \) Numeri figurati cujusvis ordinis \( n \), summa Seriei \( \frac{M}{2} + \frac{N}{4} + \frac{O}{8} + \frac{P}{16} + \&c. \) æqualis erit Numeri binari dignitati \( 2^{n-1} \). Sic Series \( \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \&c. = 2^{n-1} = 1 \), ut vulgò notum; Series \( \frac{1}{2} + \frac{2}{4} + \frac{3}{8} + \frac{4}{16} + \&c. = 2^{n-1} = 2 \); Series \( \frac{1}{2} + \frac{3}{4} + \frac{6}{8} + \frac{10}{16} + \&c. = 2^{n-1} = 4 \), & sic porrò. Scholium. Celeb. D. Jac. Bernoulli, in Tractatu suo de Seriebus infinitis, solvit illud Problema. "Invenire summam Seriei infinitæ Fractionum quarum Denominatores crescunt in Progressione quacunque Geometricâ, Numeratores vero progresiuntur vel juxta Numeros naturales, 1, 2, 3, 4, \&c. vel Trigonales 1, 3, 6, 10, \&c. vel Pyramidales 1, 4, 10, 20, \&c. aut juxta Quadratos 1, 4, 9, 16, \&c. aut Cubos 1, 8, 27, 64, \&c. eorumve multiplies." Ipsius solutionem consultat Lector. Aliam vero, & quidem multo generaliorem invenit D. Nic. Bernoulli illius Nepos, eamque (postquam ei hæc miseram, sed sine demonstratione) mecum communicare dignatus est, in epistolâ datâ 18° Septembris 1715, miris quidem inventis referentissimâ, qualibus me crebro dignatur vir Doctissimus. De hoc vero Problemate sic scribit. "Pour la somme d'un nombre determiné \( n \) de termes de la suite de votre Theoreme 7. [Corollarium primum est hujus Propositionis] j'ay trouvé cette formule \( \frac{1}{m^n} \times \) \[ \times \frac{m-1}{m-1} a + \frac{A-n}{m-1} b + \frac{B-n}{m-1} c + \frac{C-n}{m-1} d + \&c. \text{ ou les Lettres } A, B, C, \&c. \text{ marquent } \] les "los Coefficients des termes immediatement precedents. Et en mettant dans cette formule \( p + r \) pour \( n \), \( b^m \) pour \( m \), & en multipliant tout encore par \( c^{m-1} \), on a la solution de votre Prob. IX". Et me monuit Vir peritisimus hanc suam formulam generalem in nostram particularem (Cor. i. hujus propositionis) migrare quando \( n = \infty \); quippe tum evanescunt \( 1, n, n \cdot \frac{n-1}{2}, n \cdot \frac{n-1}{2} \cdot \frac{n-2}{3}, \&c \) respectu ipsorum \( m^n, A, B, C, \&c. \) adeo ut Series in eo casu sit \( \frac{a}{m-1} + \frac{A}{m-1}b + \frac{B}{m-1}c + \&c. \) quæ omnino coincidit cum nostrâ \( \frac{a}{m-1} + \frac{b}{m-1} + \frac{c}{m-1} + \&c. \) Adhuc aliam hujus Problematis solutionem, & quidem ab hisce admodum diversam, invenit D. Taylor ope Methodi suæ Incrementorum. Viri doctissimi rogatu, ad eum miseram formulam meam secundam pro solutione Problematis II di, item formulas alias spectantes ad Propositiones tertiam, quartam & quintam, sed sine demonstrationibus: quippe non dubitabam quin Vir acutissimus, atque ipse Methodi illius Incrementorum Inventor, hisce, vel saltem paribus inveniendis par esset. Rescripsit se harum solutiones invenisse, & simul alia quædam communicavit ad hujus methodi profection multum facientia, quæ jam nostro hortatu inductus hicce subjungere dignatur. Appendix APPENDIX Quâ methodo diversâ eadem materia tractatur: Auctore Brook Taylor, LL.D. R. S. Secr. Hortatu Viri Clariss. cui nos innumeris officiis de- vinctissimos esse libenter fatemur, sequentes jam Propositiones exhibemus, quas quidem in aliam occasio- nem reservandas esse decrevissemus, ni æquum visum fuisset parendum esse imperio amici qui, dum Propositio- nes quasdam præcedentes suas olim nobis investigan- das proposuit, earum inveniendarum occasionem dedit. Definitiones. 1. Quantitatis cujusvis variabilis valorem praesentem designo literâ simpliciter scriptâ, ut \( x \); valores praec- edentes distinguo lineolis eadem literæ ex parte supe- riori positis, sequentes lineolis ex parte inferiori scrip- tis. Ut vi hujus Definitionis sint \( x, x', x, x, x'' \), ejus- dem variabilis valores quinque continui, existente \( x \) va- lore praesenti, \( x' \) proximè præterito, \( x'' \) secundò præteri- to; \( x' \) proximè, atque \( x'' \) secundò futuro. Et sic de aliis. Ad eundem modum sunt interpretandæ lineolæ quæ incrementis apponuntur. Sic sunt \( x, x', x, x, x'' \), ip- sius \( x \) valores quinque continui; ut sit \( x'' \) incrementum secundum ipsum \( x \), sit \( y \) incrementum secundum ipsum \( x \). Et sic de aliis. Cor. Vi hujus Definitionis, \( x + \dot{x} = x \), \( x + x = y \), \( x + - x = x \). Et sic de aliis hujusmodi. Quando usu venit ut variabilis quantitas, puta \( x \), spectanda sit tanquam Incrementum, ejus Integrale designo literâ inter uncos [ ] inclusâ. Istimus etiam Integralis [x] integrale (vel ipsius \( x \) integrale secundum,) designo numero binario uncorum priori superimposito, ut [x]. Istimus etiam Integralis integrale (vel ipsius \( x \) integrale tertium,) ad eundem modum designo numero ternario, ut [3x]. Et sic deinceps. Unde vi hujus Definitionis constituunt [3x], [2x], [x], \( x \) Seriem terminorum, quorum quilibet est ipsum immediatè praecedentis incrementum primum, ut sit [2x] = [3x], \[ x = [2x], x = [x]. \] Lemma. Facti \( x \times v \) ex Multiplicatione duorum variabilium \( x \) & \( v \), incrementum est \( x \times v + \dot{x} \times v \). Nam autis variabilibus per propria incrementa, sit novum producendum \( x + \dot{x} \times v + \dot{v} \), sive \( x \times v + \dot{x} \times v + \dot{x} \times \dot{v} \times v \), hoc est \( x \times v + \dot{x} \times v + \dot{x} \times \dot{v} \) (pro \( x + \dot{x} \) scripto \( x \) per Def. i.) Unde dempto priori producio \( x \times v \), restat Incrementum \( x \times v + \dot{x} \times v \). Qqqqq Prop. Prop. I. Theor. Ejusdem Façii \( x \nu \) Incrementum, vel primum, vel secundum, vel tertium, vel aliud quodvis, cujus ordo designatur per symbolum \( n \), exhibetur per formulam hanc generalem \[ x\nu + n \times \frac{n-1}{2} \times \frac{n-2}{3} \times \cdots \times \frac{n-(n-1)}{n} \] In hac formulâ hæc sunt observanda, \( 1^{\text{mo}} \) Terminus sum numeri coefficientes \( 1, n, n \times \frac{n-1}{2}, n \times \frac{n-1}{2} \times \frac{n-2}{3} \cdots \). Iidem sunt ac in binomii dignitate \( n \). \( 2^{\text{do}} \) Numeri \( n, n-1, n-2, n-3, \cdots \), ipsis \( x \) infrascripti designant numeros punctorum quibus definitur Incrementa. \( 3^{\text{rd}} \) Lineolae \( \cdots \), \( \cdots \), \( \cdots \), \( \cdots \), ipsis \( x \) infrascriptae interpretandae sunt per Def. 1. \( 4^{\text{th}} \). In quovis Termino numerus punctorum ipsis \( x \) simul infrascriptorum, est \( n \). Sit \( v.g. n = 4 \): tum per formulam, ipsius \( x \nu \) incrementum quantum prodit \( x \nu + 4 \times x \nu + 6 \times x \nu + 4 \times x \nu + x \nu \). Theorema hoc generale demonstrari potest per Inductionem, incrementis continuò sumptis juxta formam in Lemmate praecedenti traditam. Sæ & collectâ formâ Series ex hujusmodi calculo, Theorema etiam demonstrari potest per Methodum Incrementorum, ad eum modum cujus specimen mox dabimus in demonstratione Propositionis tertiae. Prop. II. Theor. Ipsius \( x \nu \) Integrale primum \( [x \nu] \) exhibetur per Seriem \( [x] \nu - [x] \nu + [x] \nu - [x] \nu \cdots \). Series autem ita terminatur, ut sit \( [x \nu] = [x] \nu \). \[ - \left[ [x] v \right] = [x] v - \left[ x^2 \right] v + \left[ x^3 \right] v = 0. \] Nam sumendo incrementa restituitur propositum \( x v \). Cor. 1. Datis duobus ex istis \([x], [x v], \left[ x^2 \right] v\), datur tertium. Item datis tribus ex istis \([x], \left[ x^2 \right] v, [x v]\), \(\left[ x^3 \right] v\), datur quartum, Et sic porrò. Cor. 2. Si \( v = 0 \), datur \([x v]\) ex dato \([x]\). Si \( v = 0 \) datur \([x v]\) ex datis duobus \([x], \left[ x^2 \right] v\), Si \( v = 0 \), datur \([x v]\), ex datis tribus \([x], \left[ x^2 \right] v, \left[ x^3 \right] v\). Et sic porrò. Ex. 1. Sit exemplum hujus formulæ in inventione Integralis ipsius \(\frac{v}{z z z z}\), dato nempe \(z\), atque existente \(v = 0\), qui casus est specialis Propositionis secundæ Tractatus præcedentis Dni Monmort. Facio itaque \(x = \frac{1}{z z z z}\), sunt \([x] = \frac{-1}{z z z z}, \left[ x^2 \right] = \frac{1}{z z z z}\), atque \(\left[ x^3 \right] = \frac{-1}{z z z z}\). Unde per formulam sit \([x v]\), hoc est \(\left[ \frac{v}{z z z z} \right] = - \frac{v}{z z z z} = \frac{-v}{z z z z} - \frac{v}{z z z z}\). Ex. Ex. 2. Sit aliud exemplum in inventione Integralis ipsius \( n^a \), ubi est \( z = 1 \), atque datur \( a \). Tum pro \( x \) sumpto \( a \), & pro \( v \) sumpto \( n \), fit \( x = a^z \); hoc est \( x = ax \), seu \( x + x = ax \), adeoque \( x = a - 1 \times \), atque \( x = \frac{x}{a - 1} \). Regrediendo itaque ad Integralia fit \[ [x] = \frac{x}{a - 1}; \text{ item } [x] = \frac{x}{a - 1}, \text{ item } [x] = \frac{x}{a - 1}; \text{ & sic porrò. Adeoque (quoniam } x = ax,) \text{ sunt } [x] = \frac{x}{a - 1}, [x] = \frac{ax}{a - 1}, [x] = \frac{a^2 x}{a - 1}, \&c. \text{ Unde per formulam prodit } [n^a] = \frac{a^n}{a - 1} - \frac{a^{n+1}}{a - 1} + \frac{a^{n+2}}{a - 1} \&c. In hoc exemplo continetur Solutio Problematis, de quo agit Daus de Monmort in Propositione nona. Coincidit autem formula cum ea quam exhibet ille in Corollario primo ejusdem Propositionis. Scholium. Possunt etiam ex hâc formulâ alii derivari valores Integralis quæsiti, pro vario modo quo interpretantur Incrementi proposici factores. Sic in exemplo secundo integrale ipsius \( n^a \) exhiberi potest per formulam \( a^z[n] - a - 1 a^z[n] + a - 1^2 a^z[n] \&c. \) pro \( x \) nempe sumpto \( n \), & pro \( v \) sumpto \( a \). Sed de his fortasse alia occasione fusius dicemus. Prop. III. Theor. Ejusdem \( xv \) Integrale, vel primum, vel secundum, vel tertium, vel aliud quodvis cujus ordo designatur symbolo \( n \), exhibetur per Scriem in hâc formâ generali prodeuntem \( [xv] = [x]v - n[x]v + \cdots \). \[ n \times \frac{n+1}{2} [x] v - n \times \frac{n+1}{2} \times \frac{n+2}{3} [x] v + \ldots \] Collectâ formâ Seriei ex Propositione præcedenti, Coefficientes \( r, -n, n \times \frac{n+1}{2}, -n \times \frac{n+1}{2} \times \frac{n+2}{3}, \ldots \) sic inveniuntur per Methodum Incrementorum. Pone \[ [x v] = A[x] v + B[x] v + C[x] v + D[x] v + \ldots \] Tum aucto \( n \) incremento suo \( n = 1 \), atque ipsis \( A, B, C, D, \ldots \) incrementis suis contemporaneis \( A, B, C, D, \ldots \) ut jam evadant \( n, A, B, C, D, \ldots \) fiet novum Integrale (quod Integrale est ipsius \( [x v] \)) \( [x v] = \) \[ A[x] v + B[x] v + C[x] v + D[x] v + \ldots \] Hujus itaque Incrementum primum coincidere debet cum Integrali prius posito. Sumptis ergo incrementis, fit \[ [x v] = A[x] v + B[x] v + C[x] v + D[x] v + \ldots \] idem ac Integrale prius positum. Itaque terminos homologos inter se comparando fit \( 1^{mo} A = A \). Unde est \( A \) datum quid. Sed ubi \( n = 0 \), est \( A = 1 \), ergo \( A = 1. 2^{do}. B = B + A \), hoc est \( B = B + B + 1 \), seu \( B = -1 = -n \). Ergo regrediendo ad Integralia, fit \( B = -n + a \). Sed ubi \( n = 0 \), est \( B = 0 \). Ergo \( a = 0 \), atque \( B = -n. 3^{ro}. C = C + B \), hoc est \( C = n \). Regre- \[ K r r r r \] diendo diendo itaque ad Integralia fit $C = \frac{n^2}{2} + b$. Sed ubi $n = 0$, est $C = 0$. Ergo $b = 0$, atque $C = \frac{n^2}{2}$, hoc est, $n \times \frac{n+1}{2}$. Ad eundem modum invenitur $D = -n$ $\times \frac{n+1}{2} \times \frac{n+2}{3}$. Et sic pergendo inveniuntur cæteri Coefficientes. Scholium. 1. In hâc Propositione comparatâ cum Propositione primâ, cernitur singularis quædam relatio Incrementa inter & Integralia. Ut enim in Arithme- ticâ vulgari, Multiplicatio & Divisio sunt invicem ita contrariæ ut si Multiplicatio designetur per Indicem affirmativum, Divisio designabitur per Indicem cum signo negativo; sic etiam in Methodo Incrementorum, si Incrementum designetur per Indicem affirmativum, Index negativus Integrale sîstet. Sic in Propositione primâ, si pro $n$ sumatur Numerus binarius 2, per for- mulam exhibebitur ipsius $xv$ incrementum secundum, nempe $xv + 2xv + xv$; Sed si pro $n$ sumatur nume- rus negativus — 2, ut jam quaeratur ipsius $xv$ incre- mentum (ita loqui liceat) negativè secundum, (quod idem est ac Integrale secundum) prodeunt coefficien- tes iudem ac si sumatur $n$ affirmativè in Propositione præsenti: atque interpretatis insuper ipsis $x$, $x$, $x$, &c. per $[x]$, $[x]$, $[x]$, &c. Series fit omnino eadem ac per Propositionem præsentem prodit, ubi quaeritur In- tegrale secundum. 2. Ex his autem formulis quasi suâ sponte proce- dunt formule Propositionum undecimæ atque duode- cimæ Libri de Methodo Incrementorum. Nam pro incrementis scribe Fluxiones, atque evanescentibus incremenris fiant jam omnes \( x, x', x'', x''', \ldots \) inter se æquales, atque migrabit latinum hæc Propositio secunda in illam undecimam, atque praesens tertia in illam quodecimam. Quod quidem exemplum facis intinge eit Methodi Newtonianæ, quâ colligat ille rationes Fluxionum ex rationibus ultimis incrementorum evanescentium, vel ex primis nascentium. --- Additamentum. Præcedentium impressioni intentus dum Typothetarum erroribus corrigendis do operam, atque eà occasione in animo illa sœpius revolvo, subiit Artificium illud quo jam olim usus est D. Jac Bernoulli in inventione quarundam Serierum, opè Progressionis Harmonicæ cujus meminit D. de Monmort in Scholio 6. Prop. V. præcedente commodè etiam applicari posse ad inventionem ipsius Monmortii Propositionum 2\textsuperscript{de}, 3\textsuperscript{tæ}, 4\textsuperscript{ta}, 5\textsuperscript{ta}, atque id genus aliarum aliquanto fortasse generaliorum. Hoc in sequentibus paucis ostendisse, credebam Lectori non fore ingratum. --- Theorem. Sit Progressio Arithmetica \( p, p + n, p + 2n, \ldots \), cujus termini singuli successivè designentur per \( x, \ldots \) functo \( b, c, d, \ldots \) quivis multiplices differentiae datæ \( n \) terminorum Progressionis illius Arithmetici. Sint \( A, B, C, D, \ldots \). Numeri quilibet dati, & constituantur fractiones quovis \( \frac{A}{x}, \frac{B}{x + b}, \frac{C}{x + c}, \frac{D}{x + d}, \ldots \). Pro \( x \) successivè scriptis valoribus suis \( p, p + n, p + 2n, \ldots \). ex harum fractionum quâlibet, oritur Series Harmonicè proportionalium Sic \( v g \). ex fractione primâ \( \frac{A}{p} \), oritur Series \( \frac{A}{p}, \frac{A}{p+n}, \frac{A}{p+2n}, \&c. \). Dico quod aggregatum quotlibet hujusmodi Serierum in infinitum continuatarum in terminis numero finitis exhiberi potest, si modo fuerit numeratorum \( A, B, C, D, \&c. \) aggregatum æquale nihilo. Duobus exemplis hoc fiet manifestum. Ex. Sint duæ tantùm fractiones \( \frac{A}{x}, \) atque \( \frac{-A}{x+3n} \), existente \( b = 3n \). Scribantur Series harmonicae ex his formulis ortæ, eo ordine, ut termini, in quibus sunt denominatores æquales, sibi invicem respondeant, & collectis summis terminorum homologorum, prodibit aggregatum Serierum in terminis numero finitis, ut in calculo apposito videre est. \[ \frac{A}{p} + \frac{A}{p+n} + \frac{A}{p+2n} + \frac{A}{p+3n} + \frac{A}{p+4n} + \&c. = \text{Seriei ortæ ex } \frac{A}{x} \] \[ + \frac{-A}{p+3n} + \frac{-A}{p+4n} + \&c. = \text{Seriei ex } \frac{-A}{x+3n} \] \[ \frac{A}{p} + \frac{A}{p+n} + \frac{A}{p+2n} + \&c. = \text{Aggreg. Serierû.} \] Ex. 2. Sint tres fractiones \( \frac{A}{x}, \frac{B}{x+2n}, \frac{C}{x+3a} \), existentibus \( b = 2n, c = 3n \), atque \( A + B + C = 0 \). In hoc casu Calculus sic te habet. \[ \frac{A}{p} + \frac{A}{p+n} + \frac{A}{p+2n} + \frac{A}{p+3n} + \&c. = \text{Seriei ortæ ex } \frac{A}{x} \] \[ + \frac{B}{p+2n} + \frac{B}{p+3n} + \&c. = \text{Seriei ex } \frac{B}{x+2n} \] \[ + \frac{C}{p+3n} + \&c. = \text{Seriei ex } \frac{C}{x+3n} \] \[ \frac{A}{p} + \frac{A}{p+n} + \frac{A+B+C}{p+2n} + \&c. = \text{Aggregato Serierum.} \] Ubi Ubi etiam prodit aggregatum Serierum in terminis numero finitis, nempe \( \frac{A}{p} + \frac{A}{p+n} + \frac{A+B}{p+2n} \), ob Numeratorum \( A, B, C \), aggregatum æquale nihilo. Et ad eundem modum demonstratur Theorema in aliis casibus quibusvis. Cor. 1. Ex his principiis derivari possunt innumeræ Series in infinitum continuatæ, in terminis tamen numero finitis summabiles. Caf. 1. Sint \( \frac{A}{x} \) & \( \frac{A}{x+b} \) formulæ duarum Serierum harmonicarum quarum aggregatum prodit in terminis numero finitis per superius demonstrata, Tum, formulis istis in unam summam collectis, fit \( \frac{Ab}{x \times x + b} \) formula Seriei summabilis. Sint v.gr. \( A = \frac{1}{6}, p = 1, n = 2, \) atque \( b = 3n = 6 \). Tum formulæ Serierum harmonicarum erunt \( \frac{1}{6x} \), & \( \frac{-1}{6 \times x + 6} \), formula Seriei compositæ summabilis erit \( \frac{1}{x \times x + 6} \), Serie illa existente \( \frac{1}{1 \times 7} + \frac{1}{3 \times 9} + \frac{1}{5 \times 11} + \frac{1}{7 \times 13} + \&c. \), atque summa Seriei, per calculum in præmissis demonstratum, erit \( \frac{1}{6 \times 1} + \frac{1}{6 \times 3} + \frac{1}{6 \times 5} \). Sint tres formulæ Serierum harmonicarum \( \frac{A}{x}, \frac{B}{x+b}, \frac{C}{x+c} \), (existente \( A + B + C = 0 \), ut sit Serierum aggregatum finitum per præmissa.) Tum formulis in unam summam collectis fit \[ \frac{A \times x + b \times x + c + B \times x \times x + c + C \times x \times x + b}{x \times x + b \times x + c}, \] seu (terminis revocatis ad formam factorum \( x, x \times x + b, x \times x + b \times x + c, \)) \[ \frac{Ac + Ac + c - bB \times x + A + B + C \times x \times x + b}{x \times x + b \times x + c}, \] hoc est \[(ob \ A + B + C = 0) \frac{Acb + Ac + B \times c - b \times x}{x \times x + b \times x + c}, \text{ formula Seriei summabilis. Si quatuor sint Fractiones} \\ \frac{A}{x}, \frac{B}{x + b}, \frac{C}{x + c}, \frac{D}{x + d}, (\text{existente } A + B + C + D = 0) \\ \text{ad eundem modum invenietur formula Seriei summabilis} \\ \frac{Abcd + Acd + B \times c - b \times d - b \times x + Ad + B \times d - b \times c \times d - c \times x \times x + b}{x \times x + b \times x + c \times x + d} \\ \text{Et sic pergere licet ad formulas adhuc magis compositas.} \text{Cas. 2. Et si plures sint formulæ Serierum hujusmodi summabilium, quarum denominatorum factores excerptantur ex diversis progressionibus Arithmeticis, existarum formularum quotvis in unam summam additione, conficietur formula nova Seriei summabilis:} \\ \text{Sint e.g. formulæ duæ Serierum summabilium} \\ \frac{1}{x \times x + 3} \\ \& \frac{1}{x \times x + 2}, \text{ excerptis } x \text{ ex Progressione Arithmetica } 1, 2, 3, 4, \&c. z \text{ ex Progressione Arithmetica } 1, 3, 5, \&c. \text{ Tum ex his formulis in unam summam collectis fiet formula nova} \\ \frac{z \times x + 2 + x \times x + 3}{x \times x + 3 \times x + 2}, \text{ vel, (exposito } z \text{ per } x \& \text{ numeros datos)} \\ \frac{2 \times x - 1 \times z \times x + 1 + z \times x + 3}{x \times x + 3 \times 2 \times x - 1 \times 2 \times x + 1} \text{Cor. 2. Hinc omnis Series in infinitum continuata summabilis est, cujus termini designantur per Fractionem, cujus denominatoris factores excerptuntur ex data quâlibet Progressione Arithmetica, numerator autem est multinomium, cujus dimensiones sunt ad minimum binario pauciores, quam sunt dimensiones Denominatoris. Nam omnis hujusmodi fractio resolvi potest in tot fractiones simplices, quot sunt dimensiones (hoc est, quot sunt factores) Denominatoris, quarum numeratorum aggregatum est nihil. Sit exempli gratia, formula}\] formula oblatà \( \frac{\alpha + \beta x + \gamma x \times x + b}{x \times x + b \times x + c \times x + d} \). Pone hanc formulam aquari aggregato fractionum \( \frac{A}{x} + \frac{B}{x + b} + \frac{C}{x + c} + \frac{D}{x + d} \). Tum fractionibus istis in unam summam collectis fiet \[ \frac{A b c d}{x \times x + b \times x + c \times x + d} = \frac{\alpha + \beta x + \gamma x \times x + b}{x \times x + b \times x + c \times x + d} \] Unde per comparationem terminorum homologorum fit \( A b c d = \alpha \), \( A c d - B \times c - b \times d - b = \beta \), \( A d - B \times d - b + C \times d - c = \gamma \) \( A + B + C + D = 0 \). adeoque \( A = \frac{x}{b c d} \), \( B = \frac{\beta - A c d}{c - b \times d - b} \), \( C = \frac{\gamma - A d - B \times d - b}{d - c} \), \( D = -A - B - C \), Quo pacto formula oblatà resolvitur in fractiones simplices \[ \frac{\alpha}{b c d x} + \frac{\beta - A c d}{c - b \times d - b \times x + b} + \frac{\gamma - A d - B \times d - b}{d - c \times x + c} + \frac{A - B - C}{x + d} \] ex quibus ortarum Serierum aggregatum, hoc est, summa seriei ortae ex formulâ oblatâ \( \frac{\alpha + \beta x + \gamma x \times x + b}{x \times x + b \times x + c \times x + d} \), per jam dicta prodit in terminis numero finitis. Quod vero dimensiones numeratòri in formulâ oblatâ, debeant esse binario ad minimum pauciores, quam sunt dimensiones Denominatoris, hinc constat quod in reductione fractionum \( \frac{A}{x}, \frac{B}{x + b}, \frac{C}{x + c}, \frac{D}{x + d} \), quilibet numerator \( A, B, C, D \), ducitur in omnes denominatores excepto uno, nempe suo; unde prodeunt Numeratoris Dimensiones unitate pauciores quam sunt dimensiones Denominatoris. Sed per æquationem $A + B + C + D = 0$ perit altissima dimensio in numeratore; Unde superflunt Numeratoris Dimensiones ad minimum binario-pauciores quam sunt dimensiones Denominatoris. Ad hoc veò Corollarium revocari possunt D. de Monmort Propositiones 2da & 5ta. Cor. 3. Item oblatâ formulâ juxta Cas. 2. Cor. 1. adhuc magis compositâ, ex iisdem principiis perspici potest an sit Series summabilis. Sint progressiones duæ Arithmeticae 1, 3, 5, &c. 2, 4, 6, &c., quorum termini homologi designentur per $x$ & $z$, & sit formula Seriei oblata $\frac{\alpha + \beta x + \gamma x^2}{x \times x + 2 \times z \times z + 2}$, vel (pro $z$ scripto $x + 1$, & factoribus Denominatoris in ordinem coactis) $\frac{\alpha + \beta x + \gamma x^2}{x \times x + 1 \times x + 2 \times x + 3}$. Pone formulam hanc æquari aggregato formularum $\frac{P}{x \times x + 2}$, $\frac{Q}{x + 1 \times x + 3}$, Serierum per superius dicta summabilium, ut (formulis his novissimis in unam summam collectis) sit $\frac{P \times x + 1 \times x + 3 + Q \times x \times x + 2}{x \times x + 1 \times x + 2 \times x + 3}$ seu $\frac{3P + 4P + 2Q + P + Qx^2}{x \times x + 1 \times x + 2 \times x + 3} = \frac{\alpha + \beta x + \gamma x^2}{x \times x + 1 \times x + 2 \times x + 3}$. Hinc comparando terminos homologos oriuntur æquationes $3P = \alpha$, $4P + 2Q = \beta$, $P + Q = \gamma$. Unde eliminatis $P$ & $Q$ per debitas operationes Analyticas, prodit æquatio $2\alpha - 3\beta + \gamma = 0$, qua definitur relatio quæ inter coefficientes $\alpha$, $\beta$, $\gamma$ intercedere debet, ut Series orta ex formulâ oblata $\frac{\alpha + \beta x + \gamma x^2}{x \times x + 1 \times x + 2 \times x + 3}$ sit sit summabilis. Ad eundem modum si formulæ oblatæ Denominatoris factores excerptantur ex tribus Progressionibus Arithmeticis, inveniuntur duæ æquationes quibus definitur relationes coefficientium Numeratoris, ut sit Series summabilis. Si quattuor sint Progressiones Arithmeticæ, Coefficientium relatio definitur per tres æquationes. Et sic porrò. Et in hujusmodi formulis ut sint Series summabiles, hæc insuper observanda sunt, Primò ut Numeratorum dimensiones sint ad minimum binario pauciores quam sunt dimensiones Denominatorum, Deinde ut ex singulis Progressionibus Arithmeticis excerptantur ad minimum duo factores Denominatoris. Denique, quod si sint duo vel plures factores Denominatoris inter se æquales, ponendum sit tot etiam Progressiones Arithmeticæ, ex quibus excerptur, esse inter se æquales. Præmissis attentius perspensis, hæc obvia erunt. Ad hoc vero Corollarium facile revocantur D. de Monmort Propositiones 3ta & 4ta. FINIS. ERRATUM in N°. 352. Page 586, after the end of line 15, add black Cloud, from behind which there issued a. LONDON: Printed by W. and J. Innys, Printers to the Royal Society, at the Princes-Arms in St. Paul's Church-Yard. 1717.