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Possible Duplicate:
Computing $\zeta(6)=\sum\limits_{k=1}^\infty \frac1{k^6}$ with Fourier series.

What function do I pick for the summation from $$\sum_{n =1}^{\infty}\frac{1}{n^6} \ ?$$ using Parseval's identity

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    i dont know what that is?2012-04-27
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    The [Riemann Zeta function](http://en.wikipedia.org/wiki/Riemann_zeta_function) $\zeta(s)=\sum_1^{\infty}n^{-s}$ generalizes your sum from reciprocal powers of $6$ to (almost) any (complex) exponent $s$ (the real part of $s$ must be greater than one for the series to converge). It is an extensively and actively studied function, quite mysterious and beloved by mathematicians.2012-04-27
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    @bgins The somewhat beautiful thing with Parseval here is that it gives an easy way to actually calculate $\zeta(2n)$ $n\ge1$! Other values are much harder: (1) [Apéry]:(http://en.wikipedia.org/wiki/Ap%C3%A9ry%27s_constant) proved $\zeta(3)$ is irrational in 1979. (2) [Zudilin]:(http://en.wikipedia.org/wiki/Wadim_Zudilin) proved that $\zeta(5),\,\zeta(7),\,\zeta(9),\,\zeta(11)$ is irrational in 2001.2012-04-27

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