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I was doing my homework assignment and I did this question correctly. However, I'm interested in knowing the reason behind the logic that how $1/1^4 + 1/2^4 + 1/3^4$ .... to infinity evaluates to $pi^4/90$

Also, I know that whenever there is pi, somehow everything is related to circle. How is this question related to circle?

Here's a screenshot of my question if that helps! Thanks!

enter image description here

mehulmpt
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    This series can be derived by using Eulers Product representation of the sine function. And the Taylor series of the sine function. In general $$1+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\frac{1}{4^{2n}}+\cdots = A\cdot\pi^{2n}$$ Where $A$ is a rational number, which is related to the Bernoulli numbers. Here is a video series (https://goo.gl/0oENyJ). – MrYouMath Oct 10 '15 at 15:59
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    The formula here is one of several special values of the Riemann zeta function, $\zeta(4) = \frac{\pi^4}{90}$. The Riemann zeta function at $2n$ is $(2\pi)^{2n}$ times a rational number connected with the Bernoulli numbers. – hardmath Oct 10 '15 at 16:00
  • Same question as:http://math.stackexchange.com/questions/28329/nice-proofs-of-zeta4-pi4-90 – NoChance Oct 10 '15 at 17:43

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