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I got this question when I was doing some exercises. I was ask to establish $$ \sum_{n=0}^{\infty}\frac {1}{(2n+1)^2}=\frac{\pi^4}{96},\quad \sum_{n=0}^{\infty}\frac {1}{(2n+1)^6}=\frac{\pi^6}{960},\quad \sum_{n=1}^{\infty}\frac {1}{(n)^6}=\frac{\pi^6}{945} $$

And I was guided to evaluate some functions' Fourier series, and use Parseval Identity.

However I was wondering how they can determine which function they should use in order to evaluate the sum. For the first one, it uses $f(x)=|x|$, for the second one, it uses the odd function on $[-\pi,\pi]$, where$f(x)=x(\pi-x)$ on $[0,\pi]$. For the third one, it uses $f(x)=x^2$ and integrates it's Fourier representation before using Parseval Identity.

Is there any rule about which function I should select in order to evaluate the sum? Without the hints of the questions, I have no idea about how to solve the problems.

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1 Answer 1

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As far as I can tell there is no general rule. You try with $f(x)=A\,x^2+B\,x+C$ on $[-\pi,\pi]$, or on $[0,\pi]$ and extend to $[-\pi,\pi]$ as an even or as an odd function, and chose $A,B,C$ to get the desired result. Of course this can be done with polynomials of higher degree.

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