# Convergent Trigonometric Series is Periodic

## Theorem

Let $S \left({x}\right)$ be a trigonometric series:

$S \left({x}\right) = \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty \left({a_n \cos n x + b_n \sin n x}\right)$

Let $S$ be convergent.

Then $S$ is periodic:

$\forall r \in \Z: S \left({x + 2 r \pi}\right) = S \left({x}\right)$

## Proof

Let $S \left({x}\right)$ converge to some $L \in \R$.

Let $r \in \Z$ be arbitrary.

Then:

 $\displaystyle S \left({x + 2 r \pi}\right)$ $=$ $\displaystyle \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty \left({a_n \cos n \left({x + 2 r \pi}\right) + b_n \sin n \left({x + 2 r \pi}\right)}\right)$ Definition of $S \left({x + 2 r \pi}\right)$ $\displaystyle$ $=$ $\displaystyle \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty \left({a_n \cos \left({n x + 2 r n \pi}\right) + b_n \sin \left({n x + 2 r n \pi}\right)}\right)$ $\displaystyle$ $=$ $\displaystyle \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty \left({a_n \cos \left({n x}\right) + b_n \sin \left({n x}\right)}\right)$ Sine and Cosine are Periodic on Reals $\displaystyle$ $=$ $\displaystyle S \left({x}\right)$ Definition of $S \left({x}\right)$

$\blacksquare$