# If 1/1^2 + 1/2^2 + 1/3^2 + ........ upto infinity = pi^/6, then 1/1^2 + 1/3^2 + 1/5^2 +..... =

rajat Bansal
20 Points
8 years ago
write the question like this, divide in odd terms n even terms

1/1^2 +1/3^2 + 1/5^2.......upto infinity   +  1/2^2 +1/4^2 +1/6^2.......upto infinity = pi/6
Now from the even terms take common 1/2^2

1/1^2 +1/3^2 +1/5^2........    + 1/2^2  ( 1/1^2 + 1/3^2 + 1/5^2+................) =pi/6

now let 1/1^2 + 1/3^2 + 1/5^2..........    =K

so we can write
K +1/2^2(K) = pi/6
K +K/4 = pi/6
5*K/4= pi/6
K=2*pi/15
So
1/1^2 + 1/3^2 + 1/5^2.................. = 2*pi/15
Rishi Sharma
askIITians Faculty 646 Points
3 years ago
Dear Student,
Please find below the solution to your problem.

1/1^2 +1/3^2 + 1/5^2.......upto infinity + 1/2^2 +1/4^2 +1/6^2.......upto infinity = pi/6
Now from the even terms take common 1/2^2
1/1^2 +1/3^2 +1/5^2........ + 1/2^2 ( 1/1^2 + 1/3^2 + 1/5^2+................) =pi/6
now let 1/1^2 + 1/3^2 + 1/5^2.......... =K
so we can write
K +1/2^2(K) = pi/6
K +K/4 = pi/6
5*K/4= pi/6
K=2*pi/15
So
1/1^2 + 1/3^2 + 1/5^2.................. = 2*pi/15

Thanks and Regards