VOL. LXXXI.] PHILOSOPHICAL TRANSACTIONS. 81 



Exam. 3. Let the genera! term be y-^ = i + i + \ + &c., and, by writ- 



ing 1, 3, 4, &c. for x, we have 7 + ^+55 -f-&c. = b + e + h-|- &c. = 

 •■221689395104. 



Exam. 4. Let the general term be g^, _^ = 3? + 9? + 27?"^ "^ ^^'' ^"^' 



by writing 1, 3, 4, &c. for x, &c. we have^g + 2Ji + 7g6"*'*^'^-=^5'^ + i° + ^ 

 L + &c. = .028385252052. 



Exam. 5. To find the sum of the series g — ^ + ^3 — 75;^ + &c. If we 



write 2, — 3, 4, — 5, &c. for r, the general term will be -j-— - = — — -^-\- - 



\^-\- &c. Now, by writing 2, — 3, 4, — 5, &c. for x, the serieses of which 



i -, &c. are the general terms, will be alternately + and — , and therefore their 



sums will be found in tab. 2, and the serieses of which -5, -fj, &c. are the general 

 terms will have their terms all +, and therefore their sums will be found in tab. 

 1 . Hence the sum required = b -{- k -^ -\- &c. = — e — l— r — &c. = 

 .082800931803. 



Prop. 3. — To find the sum of the sums of the reciprocals of the odd powers 

 in tab. 2. 



By division -^^^^rVT^Tx = i^ + I + i + ^^ + ^^^ '^^"'^^ ^^ ^"''"S 2, - 3, 

 4, — 5, &c. for X, the sums of the serieses of which — , -, &c. are the general 

 terms, may be found by tab. 1, and the other sums by tab. 1 ; hence -— - + - — 

 + ^ + &c. = A + c + E + &c. + i + c^ + / + &c. ; but j-i^ + ^ + ^ 



I 3 



+ &c. = — - + 2 hyp. log. 2; and by prop. 1, a + c + e + &c. = - ; hence 

 b^d-^f+hc. = -\+1 hyp. log. 2. 



Prop. 4 — To find the sum of the infinite series whose general term is — rqr-» 



By division ,.' = — — + ~- — I — -— + &c. ad inf. ; hence the sum of 



x^ , 



the series of which — ,. ^ is the general term, is found as in prop. 2. Here r 



nuist be greater than s at least by 2, otherwise the sum will be infinite. 



J.1 111 

 Exam. 1. Let the general term be , , ■ = — -.■\ — rr — &c.; hence if 



° x' + \ X' x^ x^' 



4916 



for X we write 2, 3, 4, &c. we have — + r;; + ^ + ^c. = a — b + i— w-f- 



&c. = .538527924723. — If for x we write 2, 4, 6, &c. we get ^ + ^ + -^. 

 -f &c. = a" — e" + i" — n" + &c. = .396257616555. 



VOL. XVII. M 



