84 PHILOSOPHICAL TRANSACTIONS. [aNNO IJQl. 



Ex. I. To find the sum of j-^ + ^-^-^ + j~-^ + &c. Here the 



general term js (, _ ,) ^ ,\ ^ (, + ,) = ^^^ZTT) = JTZl " P ' "°^^ ^ 

 writing 2, 3, 4, &c. for ^, the sum of the series whose general term is — is 



3 ' . 3 



:= -, by prop. 2; hence the sum required = - — A. 



Ex. 1. Let the given series be - — -5—- + . + ^ + &c. Here the 

 general term is the same as before, writing 2, 4, 6, &c. for x; and, by prop. 

 2, the sum of the series whose general term is is = -; hence the sum = 



H" 



Ex. 3. In like manner the sum of the series — —^ — - -{ — -| - + 



&C. = .221689395104 — B. 



Ex. 4. To find the sum of --^ + —^1— + j^r^jz + &c. Here the 

 general term is (,._ ^ ^ ,\ ^ (,x + ^ = ,. ^ ',. _ ,^ = ^, - \^ but thesum 



of the series whose general term is is = .O86662976264; hence the sum 



required = .O86662976264 — c. 



Prop. 5. To find the sum of the infinite series T7+75 + ^4-j5g + ^ + 



&c. 



In this series the 4th diflferences of the denominators = O; therefore the ge- 

 neral term must be represented by —, p-— --j; write therefore 2, 3, 4, &c. 



for 37, and we have %a -\- 4b -\- 1c -\- d ■= 15, 27a + 9Z; + 3c + <f = 40, 64a + 

 166 + 4c + rf = 85, 125a -f- 25^'+ 5c + rf = 156; hence a= \,b = l,c=l, 



x/= 1, and the general term is ^rqr^i^pr:;-! = i -!"+!"- Jx + ^^-'^ ^ence 

 the sum = b — c + f— G-fK — l+&c. = .1242700165. 



2 3 4 



Prop. 6. To find the sum o/" 52 + 7^2 + j^i + &c. ad inf. 



The general term = ^—^ = J, " I - I + &c.; hence, by writing 2, 3, 

 4, &c. for X, we have the sum = b — 2d + 3f — &c. = .147115771^69. 

 In like manner |, + | + ^, + &c. = b + 2d + 3f + &c. = .3 12498999865. 



Prop. 7- To find the sum of -,-^- + g^-^, + ~-^ + &c. ad ijf. 



The general term is ^r -r. . ^ , , , = t + -7^ + — + ^^•' hence, by 



writing 2, 3, 4, &c. for x, we have the sum = g + ^l + 3p + &c. = 

 .009447690684. 



Prop. 8. To find the sum of ^, J^ — , + ^^ ^ \, —, + 33 \, -3- +&c. acf in/. 



