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

## 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 the series:

$\displaystyle \sum_{n \mathop = 1}^\infty \left({\left\lvert{a_n}\right\rvert + \left\lvert{b_n}\right\rvert}\right)$

be convergent.

Then $S$ is a convergent series.