Convergent Complex Series/Examples/((-1)^n + i cos n theta) over n^2/Proof 2

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 $\text{(i)}$