VOL. LXXXI.J PHILOSOPHICAL TRANSACTIONS. 83 



H + &c. = (by prop. 1) ^ - B. Hence also — — + — '— + &c.-= ^ ■- b 

 — D ; and so on as before. 

 If the general term be under the form —— - — ; — -, it will be most convenient to 



resolve it thus: by division— = ^ - J + ^ - &c. + ^^—^y ^^"^^ ± 



J (_} i J. ^ _ ^' + &c.) X - = ( - h - _ - 4- 



x° . (x + m) V + m .r ^^ x' r' ^ ' m' ^ x . (x + m) ' x' x' "^ 



&c.) X — , where the sign on the left hand will be + or — according as n is 

 even or odd, and the number of terms on the right is = n. Now the sum of 

 the series whose general term is — -— — r is well known, and the sums of the 



«• ■ iX ^p 7/1} 



other are found from the tables. 



Exam. 1. To find the sum of-j- - + --- + — — - + &c. ad inf. Here the 



general term is ^, ^ ^^^ ^ ^^ = - ^t^^^T) + P ^^^ ^^ '"""^'"S 2, 3, 4, &c. for 



X, we have the sum = — 2~3 ~ 3~i~ ^'^' + ■'^ = ~ 2 "^ ■'^* ^" ^^^^ manner 



W:-s + ir-,+ 6^+^-- = -^+ hyp. log. 2 + A". Also ^- + ^-^ + 



If TO be nesrative, then — — : = ( — -, r , r — &c.) X — ,. Hence 



' X" . (x — m) ^x . (x — »i) x' X' ' m" 



~ 1 — ■^- -I + &c. = 1 — A — B— c; and so on for others of the 



2 



same kind. 



If the general term be under this form ^ — ■, then, in like manner, we 



have + - — ;— — — r = (— -— + -rr: — &c.) X — ;, where the sign on 



— j'" . {ax" + "0 ^ax" + m ax" ' a'.i*" ' m' ' ° 



the left hand will be + or — j according as r is even or odd, and the number of 

 terms on the right is = r + 1 • 



Exam. 1. To find the sum of — — - + „, ,„ A — ; + &c. Here m =. \, 



S'f.s ' S''. 10 ' 4-'. 17 ' 



72 = 2, r = 2, a = 1, and the general term -2 , , , . - = -I — ; 



' ' ' ° I* X (x» +1) x» + 1 X^ ^ X'*' 



now the sum of the series whose general term is - ^ is ^ .576674037469, by 



prop. 1; consequently the sum required = .576674037469 — A + c = 

 .014063204332. 



Exam. 1. If the given series be - + — — + -r^-rj + &c. the general term 



will be ^ = — —rr\ + ^; hence, by writing 2, 3, 4, &c. for x, we 



have the sum = — .576674037469 + a = .06826002938. 



If m be negative, then ,„ , „ = {-^ ; ^ — &c.) X —,. 



° ' x" . {ax" — in) ^ax" — m ax" aV" ' m' 



M 2 



