# Prove the following series

## Homework Statement

Prove that, by putting x=0 x=∏ in $x^{2}=\frac{\pi^{2}}{3}+4 \sum\limits_{n=1}^\infty \frac{1}{n^{2}} cos(nx) cos(n \pi)$, that $\frac{\pi^{2}}{8}= \sum\limits_{n=1}^\infty \frac{1}{(2n+1)^{2}}$

## The Attempt at a Solution This a solved problem, I've understood the first two parts, and how the even elements of the series were eliminated, but what about $\frac{\pi^{2}}{6}$ and $\frac{\pi^{2}}{8}$?

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## Homework Statement

Prove that, by putting x=0 x=∏ in $x^{2}=\frac{\pi^{2}}{3}+4 \sum\limits_{n=1}^\infty \frac{1}{n^{2}} cos(nx) cos(n \pi)$, that $\frac{\pi^{2}}{8}= \sum\limits_{n=1}^\infty \frac{1}{(2n+1)^{2}}$

## The Attempt at a Solution

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This a solved problem, I've understood the first two parts, and how the even elements of the series were eliminated, but what about $\frac{\pi^{2}}{6}$ and $\frac{\pi^{2}}{8}$?

Rewrite series 2 in the following way:
$$\frac{\pi^2}{3}=4\left(\left(\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\cdots \right)-\left(\frac{1}{2^2}+\frac{1}{4^2}+\frac{1}{6^2}+\cdots \right)\right)$$
$$\Rightarrow \frac{\pi^2}{3}=4\left(1+\sum_{n=1}^{\infty} \frac{1}{(2n+1)^2} -\frac{1}{2^2}\left(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots \right)\right)\,\,\,\,\,\,(*)$$
From (3), you have ##\frac{\pi^2}{6}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots##, put this in ##(*)## and you should be able to reach the correct answer.

You can turn the equation (2) into an expression for ##\frac{\pi^2}{12}## easily enough, and then ##\frac{\pi^2}{12}+\frac{\pi^2}{6}=\dots##

(and I disagree with Pranav-Arora - ##\frac{\pi^2}{8}## is correct)

Edit - Ah I see now what Pranav-Arora meant - the final summation should run from n=0, otherwise, indeed, the LHS should be ##\frac{\pi^2}{8}-1##

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