Fourier Series/Fourth Power of x over Minus Pi to Pi

Theorem
For $x \in \openint {-\pi} \pi$:
 * $\displaystyle x^4 = \frac {\pi^4} 5 + \sum_{n \mathop = 1}^\infty \frac {8 n^2 \pi^2 - 48} {n^4} \cos n \pi \cos n x$

Proof
Since $x^4 = \paren {-x}^4$, $x^4$ is an even function.

By Fourier Series for Even Function over Symmetric Range, the Fourier series of $\map f x$ can be expressed as:


 * $x^4 \sim \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty a_n \cos n x$

where for all $n \in \Z_{> 0}$:

This gives:


 * $x^4 \sim \dfrac {\pi^4} 5 + \displaystyle \sum_{n \mathop = 1}^\infty \frac {8 n^2 \pi^2 - 48} {n^4} \cos n \pi \cos n x$