Trigonometric Series is Convergent if Sum of Absolute Values of Coefficients is Convergent

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


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

Let the series:
 * $\ds \sum_{n \mathop = 1}^\infty \paren {\size {a_n} + \size {b_n} }$

be convergent.

Then $\map S x$ is a convergent series for each $x \in \R$.

Proof
For all $n \in \N_{\ge 1}$ and $x \in \R$, we have:

By hypothesis, the series $\ds \sum_{n \mathop = 1}^\infty \paren {\size {a_n} + \size {b_n} }$ is convergent.

By the Comparison Test, it follows that:
 * $\ds \sum_{n \mathop = 1}^\infty \paren {\size {a_n} + \size {b_n} }$

is absolutely convergent for all $x \in \R$.

From Absolutely Convergent Real Series is Convergent, it follows that $\map S x$ is convergent for all $x \in \R$.