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### Riemann zeta function at odd positive integers

Starting with the famous Basel problem, Euler evaluated the Riemann zeta function for all even positive integers and the result is a compact expression involving Bernoulli numbers. However, the ...
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### Various evaluations of the series $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^3}$

I recently ran into this series: $$\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^3}$$ Of course this is just a special case of the Beta Dirichlet Function , for $s=3$. I had given the following solution:...
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### Elementary method to compute $\zeta (3)$

What is an elementary method to compute $\zeta (3)$ which can be understood by a high school student? I know how to compute $\zeta (2)$ but not $\zeta (3)$. Any help is very much appreciated.
This is a p-series: $$\sum_{n=1}^\infty \frac{1}{n^p}$$ There are 2 p-series (to my knowledge) that somehow reach pi: $$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$$ \sum_{n=1}^\infty \frac{...
Use the function $f(z) = \dfrac{(\log z)^2}{z^2+1} ; ( |z|>0, - \frac{- \pi}{2} < \arg(z)< \frac{3 \pi}{2})$ to show that $\int_{0}^{\infty} \dfrac{(\ln x)^2}{x^2+1}\, dx = \dfrac{\pi^3}{8}$ ...