Asked by Sean

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
Answered by MathMate
Let
S = 1/1<sup>2</sup> + 1/2<sup>2</sup> + 1/3<sup>2</sup> + 1/4<sup>2</sup> + ...

then
S<sub>1</sub>
= 1/1<sup>2</sup> + 1/3<sup>2</sup> + 1/5<sup>2</sup> + 1/7<sup>2</sup>...
= 1/1<sup>2</sup> + 1/2<sup>2</sup> + 1/3<sup>2</sup> + 1/4<sup>2</sup> + ...
- (1/2<sup>2</sup> + 1/4<sup>2</sup> + 1/6<sup>2</sup> + 1/8<sup>2</sup> + ...)
= 1/1<sup>2</sup> + 1/2<sup>2</sup> + 1/3<sup>2</sup> + 1/4<sup>2</sup> + ...
- (1/1<sup>2</sup> + 1/2<sup>2</sup> + 1/3<sup>2</sup> + 1/4<sup>2</sup> + ...)/4
= S - S/4
= (π<sup>2</sup>/6)*(1-1/4)
= π<sup>2</sup>/8
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