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:
- $\map S x = \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$
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:
\(\ds \size {a_n \cos n x + b_n \sin n x}\) | \(\le\) | \(\ds \size {a_n \cos n x} + \size {b_n \sin n x}\) | Triangle Inequality for Real Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \size {a_n} \size {\cos n x} + \size {b_n} \size {\sin n x}\) | Absolute Value Function is Completely Multiplicative | |||||||||||
\(\ds \) | \(\le\) | \(\ds \size {a_n} + \size {b_n}\) | Real Cosine Function is Bounded and Real Sine Function is Bounded |
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 $x \in \R$.
$\blacksquare$
Sources
- 1961: I.N. Sneddon: Fourier Series ... (previous) ... (next): Chapter One: $\S 1$. Trigonometrical Series