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

Theorem
For $x \in \openint {-\pi} \pi$:
 * $\displaystyle x^6 = \frac {\pi^6} 7 + \sum_{n \mathop = 1}^\infty \frac {12 n^4 \pi^4 - 240 n^2 \pi^2 + 1440} {n^6} \cos n \pi \cos n x$

Proof
Since $x^6 = \paren {-x}^6$, $x^6$ is an even function.

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


 * $x^6 \sim \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty a_n \cos n x$

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

This gives:


 * $x^6 \sim \dfrac {\pi^6} 7 + \displaystyle \sum_{n \mathop = 1}^\infty \frac {12 n^4 \pi^4 - 240 n^2 \pi^2 + 1440} {n^6} \cos n \pi \cos n x$

Also see

 * Fourier Series of Fourth Power of x