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:

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

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

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

$\ds \size {a_n \cos n x + b_n \sin n x}$

$\le$

$\ds \size {a_n \cos n x} + \size {b_n \sin n x}$

$\ds $

$=$

$\ds \size {a_n} \size {\cos n x} + \size {b_n} \size {\sin n x}$

$\ds $

$\le$

$\ds \size {a_n} + \size {b_n}$

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 $x \in \R$.

$\blacksquare$

1961: I.N. Sneddon: Fourier Series ... (previous) ... (next): Chapter One: $\S 1$. Trigonometrical Series