Convergent Trigonometric Series is Periodic

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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 $S \left({x}\right)$

$\blacksquare$


Sources