# Calculate $\sum_{n=1}^\infty \frac{1}{n^4}$. [duplicate]

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Calculate $\sum_{n=1}^\infty \frac{1}{n^4}$.

Remark: I know that $\sum_{n=1}^\infty \frac{1}{n^4}=\frac{\pi^4}{90}$, but not how to prove that, I totally stalled.

## marked as duplicate by Semiclassical, Winther, Jack D'Aurizio, JKnecht, JMoravitzMay 2 '16 at 0:38

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• – user222031 May 2 '16 at 0:21

## 1 Answer

Try using the identity $||f||^2 = \sum_1^\infty |{\hat f}(n)|^2$ where ${\hat f}(n) = \int_0^1 f(x) e^{-2\pi i n x} dx,$ and $f(x) = x^2$ and is periodic with period 1, meaning $f(x) = f(x+1).$ The norm involved would be $||f||^2 = \int_0^1 f(x)^2 dx.$ Do the details out carefully and use the fact that $\sum_1^\infty n^{-2} = \pi^2/6$ to get the result.

Its a quite beautiful way to find the sum, I thought, the first time I worked it out. Do you see how you might change the function $f(x)$ to find $\sum_1^\infty n^{-2} = \pi^2/6$ too?