VOL. LXXII.] PHILOSOPHICAL TRANSACTIONS. 30g 



XXF. A New Method of Investigating the Sums of Infinite Series. By the 

 Rev. S. Fince, A. M., of Cambridge, p. 38g. 



The subject of this paper is divided into 3 parts : the first, Mr. V. says, con- 

 tains a new and general method of finding the sum of those series which De 

 Moivre has found in one or two particular cases ; but whose method, though it 

 be in appearance general, will on trial be found to be absolutely impracticable. 

 The 2d contains the summation of certain series, the last differences of whose 

 numerators become equal to nothing. The 3d contains observations on a cor- 

 rection which is necessary in investigating the sums of certain series by collecting 

 two terms into one, with its application to a variety of cases. 



We must however content ourselves with a very abridged state of this paper ; 

 retaining only a few specimens of the several ingenious methods above-men- 

 tioned : and besides, altering a little the notation of some of the characters or 

 ligatures, for the greater ease and simplicity in printing. 



Part 1. — Lem. 1. Let r be any whole number ; then the fluent of — ~ — can 



1 + X* 



always be exhibited by circular arcs and logarithms ; but when x = 1, the fluent 

 of the same fluxion will be expressed by the infinite series 



l — — — + - r ■ — 3 t + &c. the sum of this series therefore can always 

 be found by circular arcs and logarithms. 



Lem. 2. To find the sum of the infinite series 

 _f 1+A i a + " b _ & c 



1 . r + 1 r + 1 . 2r + 1 ~ 2r + 1 . Sr + 1 



Assume 1 — — - + gp-— — 3-7^— + &c - • = s ; then by several alge- 

 braical processes, Mr. V. rinds the sum required to be as follows : viz. — — — 



1 . r + 1 

 a + b 1 a 4 26 {Ira— (r + 2) b) x s - ra + (,■ + 1) b 



r + 1 . 2r + 1 ' Qr + 1 . 3r + 1 " 7 " ~* 



Cor. 1. Hence it appears that the sum of this series can never be exhibited in 



finite terms, except a : b as r + 2 : 1r, in which case the sum is equal to — ?- 



^ r + 2 



Hence, if a = 3, b = % then r = 1 ; .-. — - -i- -f -L - & c . . . = 1. 



Ifa=1 , i= 4,thenr4;,. i i ? -- 9 n - r .^-^+&c = '. 



Ifa=i,i=8,tl«r=|;^^- n r Tf + I J^- 5 ^+ &c =±. 



Prop. 1. — To find the sum of 



m . m 4 



T . r 4 1 . 2r 4 1 r + 1 . 3r 4 

 Every series of this kind may be resolved into the following series 



m . m 4 n , m 4 2« 



1 . r 4 1 . 2r 4 1 "*" r 4 1 . 3r 4 1 . 4r 4 1 ' Ar 4 1 . or 4 1 . 6r 4 1 t" ^ c - 



