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The following is a question from a past exam in my university in a course called "Mathematical Methods for Statistics". It consists of two subquestions that may or may not be related (there is a high chance they are related based on similar exams by the same professor).

(a) Write the Fourier series for the function $f(x) := x(\pi - |x|)$ in the interval $[-\pi, \pi]$.

(b) Calculate the sum of the following series: $$ S :=\sum_{n = 0}^\infty \frac{1}{(2n + 1)^6} $$


Attempted solution

(a) $$ f(x) \sim \frac{8}{\pi} \sum_{n = 0}^\infty \frac{\sin((2n + 1) x)}{(2n + 1)^3} $$

(b) Based on past exams, I expect the result to follow from part (a) by evaluating $f(x)$ at some appropriately chosen $x$, but I can't figure out what this $x$ might be. The best I got was by assigning $x := \frac{\pi}{2}$, which yields: $$ f(\frac{\pi}{2}) = \frac{8}{\pi}\sum_{n = 0}^\infty (-1)^n \frac{1}{(2n + 1)^3} $$

If we define $a_n := \frac{(-1)^n}{(2n + 1)^3}$, we have that $S = \sum_{n = 0}^\infty a_n^2$. But how do I proceed from here?

Another possibility is that my answer to part (a) is wrong.

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    $\begingroup$ You have $(2n+1)^{-3}$. You want $(2n+1)^{-6}$. How do you get from the first to the second. Which theorem is related to that operation in the context of Fourier analysis? $\endgroup$ – Daniel Fischer Jun 23 '14 at 19:50
  • $\begingroup$ @DanielFischer: the Parseval identity? $\endgroup$ – Evan Aad Jun 23 '14 at 19:56
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    $\begingroup$ Exactly that one. $\endgroup$ – Daniel Fischer Jun 23 '14 at 19:57
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    $\begingroup$ [Here] is Parseval identity. $\endgroup$ – Mhenni Benghorbal Jun 23 '14 at 20:03
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Thanks to Daniel Fischer's help, and assuming correctness of my answer to part (a), here's my answer to part (b).

By Parseval's identity,

$$ \frac{1}{\pi} \int_{-\pi}^{\pi} x^2 (\pi - |x|)^2 dx = \frac{64}{\pi^2}\sum_{n = 0}^\infty \frac{1}{(2n + 1)^6} $$

Whence

$$ S = \frac{\pi^6}{960} $$

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