# Fourier Series/x over Minus Pi to Pi

(Redirected from Fourier Series of x)

## Theorem

For $x \in \openint {-\pi} \pi$:

$\displaystyle x = 2 \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \map \sin {n x}$

## Proof

From Odd Power is Odd Function, $x$ is a Odd Function.

$\displaystyle x \sim \sum_{n \mathop = 1}^\infty b_n \sin n x$

where:

 $\displaystyle b_n$ $=$ $\displaystyle \frac 2 \pi \int_0^\pi x \map \sin {n x} \rd x$ $\displaystyle$ $=$ $\displaystyle \frac 2 \pi \intlimits {\frac {\sin n x} {n^2} - \frac {x \cos n x} n} 0 \pi$ Primitive of $x \sin n x$, Fundamental Theorem of Calculus $\displaystyle$ $=$ $\displaystyle -\frac 2 \pi \intlimits {\frac{x \cos n x} n} 0 \pi$ Sine of Multiple of Pi $\displaystyle$ $=$ $\displaystyle -\frac 2 \pi \paren {\frac{\pi \cos n x} n - \frac {0 \cos 0} n}$ $\displaystyle$ $=$ $\displaystyle -\frac {2 \pi \cos n \pi} {\pi n}$ $\displaystyle$ $=$ $\displaystyle -\frac {2 \paren {-1}^n} n$ Cosine of Multiple of Pi $\displaystyle$ $=$ $\displaystyle \frac {2 \paren {-1}^{n + 1} } n$

Substituting for $b_n$ in $(1)$:

$\displaystyle x = 2 \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \map \sin {n x}$

as required.

$\blacksquare$