invejligaling the Sums of infinite Series. 41 y 
— ; hence the fum of the latter feries - — will be equal to the, 
2V 2V 1 
fum of the former . 
case 1 . Let the fiven feries be - — — f? -f See. 
3 5 7 9 
Here x = 3, % = 2, w = 1 , v = 1 , n — 2, 4, 6, 8, &e. Now,.,, 
if we begin to collect at the firft term, the feries becomes 
— 4- — 2 _ 4. Sc c. and the correction, to be added , being . , 
3-5 7-9 11 • 13 
we have — 2 - + — — + 
3*5 7-9 11 • !. 
4 -&C. 4- 1 for the fum of the given 
feries; but if A= a circular arc of 45° whole radius is 
unity, it is well known that 7 
r—— + — b See. = i ~ A 
S ' 5 7-9 11-13 
therefore the fum of the given feries is 2- A.. 
case 2. Let the given feries be - 1 - - — 4 3 ? _ 2d 4- Sec.. 
1234 
Here w=i, v—i, x = 1 6, z — 1 1, n — o, 1, 2, 3, &c. 
Now, if we begin to collect at the firft term, the feries becomes 
JL- 4 — _5 ^ c . and the correction, to be added, being 
1 . 2 3 . 4 5 • 6 & 
— , we have — - — \- — — |- _J_ 4- Sec. 4- — for the fum of the given . 
2 1.23.45.6 2 0 
feries ; but + f— + + Sec. is equal to 5 x hyp. log. of 2, 
1 . 2 
confequently the fum of the given feries is equal to 
iiq_ 5 x hyp. log. of 2. 
2 
Becaufe 
axv—awx 
iv-rnvx it’ 4- n-\-a . v 
, the general term of the feries 
formed 
K*'. 
