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Evaluate Riemann Zeta function at all positive even integers

Reimann zeta function is the following

\(\zeta(s) = \displaystyle\sum_{n=1}^{\infty}{\frac{1}{n^s}}\)

Euler proved the following

\(\displaystyle\sum_{n=1}^{\infty}{\frac{1}{n^2}} = \frac{\pi^2}{6}\)

 

Read this Euler's proof of the Basel Problem. Now apply Euler's proof to evaluate the Reimann zeta function at all positive even integers 4, 6, 8, 10, etc. In particular, prove the following:

\(\displaystyle\sum_{n=1}^{\infty}{\frac{1}{n^4}} = \frac{\pi^4}{90}\)

\(\displaystyle\sum_{n=1}^{\infty}{\frac{1}{n^6}} = \frac{\pi^6}{945}\)

\(\displaystyle\sum_{n=1}^{\infty}{\frac{1}{n^8}} = \frac{\pi^8}{9450}\)

\(\displaystyle\sum_{n=1}^{\infty}{\frac{1}{n^{10}}} = \frac{\pi^{10}}{93555}\)

 

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