Convergent Trigonometric Series is Periodic

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Theorem

Let $\map S x$ be a trigonometric series:

$\map S x = \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$

Let $S$ be convergent.


Then $S$ is periodic:

$\forall r \in \Z: \map S {x + 2 r \pi} = \map S x$


Proof

Let $\map S x$ converge to some $L \in \R$.

Let $r \in \Z$ be arbitrary.


Then:

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

$\blacksquare$


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