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In several courses now, I have seen the following:

[tex]\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{{\pi}^2}{6}[/tex]

and

[tex]\sum_{n=1}^{\infty} \frac{1}{n^4} = \frac{{\pi}^4}{90}[/tex]

and so forth. While I know that these are related to the Riemann Zeta Function for even powers of n, I was wondering if there was a way to analytically solve these sorts of sums, without recourse to it. Is it possible to extend Euler's method of solving the Basel Problem to higher orders of even n (expanding the sine series, collecting the roots, and equating terms)?

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# Extending the Basel Problem

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