If Σ 1/k^2 from k=1 to infinity = pi^2/6
Use that to show that Σ 1/(2k-1)^2 from k=1 to infinity = pi^2/8
Let
S = 1/12 + 1/22 + 1/32 + 1/42 + ...
then
S1
= 1/12 + 1/32 + 1/52 + 1/72...
= 1/12 + 1/22 + 1/32 + 1/42 + ...
- (1/22 + 1/42 + 1/62 + 1/82 + ...)
= 1/12 + 1/22 + 1/32 + 1/42 + ...
- (1/12 + 1/22 + 1/32 + 1/42 + ...)/4
= S - S/4
= (π2/6)*(1-1/4)
= π2/8