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How could you compute this:

$$\sum_{n}^\infty\frac{1}{\left( 2n+1\right)^4}$$

I know the solution is $\pi^4/96$, according to Maple, but I have no idea how to do this or how to start.

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Can you first show what $\sum \frac{1}{n^4}$ is? – Gautam Shenoy Apr 21 '13 at 15:36
What mathematics do you know? – Thomas Andrews Apr 21 '13 at 15:37
up vote 7 down vote accepted

At heart, this is a simple consequence of the result:

$$\frac{\pi^4}{90}=\sum_{n=1}^\infty \frac{1}{n^4}$$

Then your value is:

$$\begin{align}\sum_{n=0}^\infty \frac{1}{(2n+1)^4} &= \sum_{n=1}^\infty\frac{1}{n^4} - \sum_{n=1}^\infty\frac{1}{(2n)^4} \\&= \left(1-\frac{1}{16}\right)\sum_{n=1}^\infty \frac{1}{n^4} \end{align} $$

You might also see the answers at: Nice proofs of $\zeta(4) = \pi^4/90$? for why the first statement is true.

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