# Infinite Sum Convergence [duplicate]

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 $\frac{1}{k^4}$.)

• 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". – Franklin Pezzuti Dyer May 29 '17 at 19:29
• Keep in mind: Apéry didn't invent it, he proved it's irrational. – Professor Vector May 29 '17 at 19:30
• Now, it converges to $\zeta(4)$. – Professor Vector May 29 '17 at 19:34
• 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. – Mark Viola May 29 '17 at 19:34
• @ProfessorVector Not at all. Another way is through Fourier Series analysis and exploitation of Parseval's theorem. – Mark Viola May 29 '17 at 19:38