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

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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 the series:

$\displaystyle \sum_{n \mathop = 1}^\infty \paren {\size {a_n} + \size {b_n} }$

be convergent.


Then $S$ is a convergent series.


Proof


Sources