I have some problems which says show that(adsbygoogle = window.adsbygoogle || []).push({});

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

and

(ii) [tex] \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^2} = \frac{\pi^2}{12} [/tex]

And another one which says, show that for [tex] 0<x<\pi [/tex]

[tex] sin x + \frac{sin 3x}{3} + \frac{sin 5x}{5} + ... = \frac{\pi}{4} [/tex]

The problem is that, the function I am supposed to work with ([tex] f(x) [/tex]) is not given. So I want to know if there is some way to find out which function I should start out with to achieve the results.

If I know f(x), I can write it as a fourier series and then substitute values of x to get the numerical series. But without knowing what f(x) to start with, I am stuck.

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# Evaluation of Numerical series by Fourier series

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