Convergent Complex Series/Examples/((-1)^n + i cos n theta) over n^2/Proof 2
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Example of Convergent Complex Series
The series $\ds \sum_{n \mathop = 1}^\infty a_n$, where:
- $a_n = \dfrac {\paren {-1}^n + i \cos n \theta} {n^2}$
is convergent.
Proof
\(\ds \sum_{n \mathop = 1}^\infty \cmod {\dfrac {\paren {-1}^n + i \cos n \theta} {n^2} }\) | \(\le\) | \(\ds \sum_{n \mathop = 1}^\infty \dfrac {1 + \cmod {\cos n \theta} } {n^2}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \sum_{n \mathop = 1}^\infty \paren {\dfrac 2 {n^2} }\) |
Thus $\ds \sum_{n \mathop = 1}^\infty \paren {\dfrac {\paren {-1}^n + i \cos n \theta} {n^2} }$ is absolutely convergent.
The result follows from Absolutely Convergent Series is Convergent: Complex Numbers.
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.3$. Series: Example $\text{(i)}$