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What value does the infinite sum $\sum_{k=1}^{\infty} \frac{1}{k^4} $ converge to? I know that $\sum_{k=1}^{\infty} \frac{1}{k^2} $ converges to $\frac{\pi^2}{6}$ but I simply do not have a clue as to where to start in order to find the value to which the first sum converges. (Sorry. I had to edit the question; my problem did not involve $\frac{1}{k^3}$ but rather $\frac{1}{k^4}$.)

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    $\begingroup$ Mathematicians have yet to find a closed-form value for $\zeta(3)$. If you are interested, do some research on the Riemann Zeta function. The answer to your question is the number $\approx 1.202$, which is also called "Apéry's Constant". $\endgroup$ – Franklin Pezzuti Dyer May 29 '17 at 19:29
  • $\begingroup$ Keep in mind: Apéry didn't invent it, he proved it's irrational. $\endgroup$ – Professor Vector May 29 '17 at 19:30
  • $\begingroup$ Now, it converges to $\zeta(4)$. $\endgroup$ – Professor Vector May 29 '17 at 19:34
  • $\begingroup$ Are you familiar with contour integration and the residue theorem? If so, evaluate $\oint_{|z|=N+1/2}\frac{\cot(z)}{z^4}\,dz$ and watch the "magical" appearance of the series of interest. $\endgroup$ – Mark Viola May 29 '17 at 19:34
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    $\begingroup$ @ProfessorVector Not at all. Another way is through Fourier Series analysis and exploitation of Parseval's theorem. $\endgroup$ – Mark Viola May 29 '17 at 19:38