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

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

\(\ds \dfrac {\paren {-1}^n + i \cos n \theta} {n^2}\)

\(=\)

\(\ds \dfrac {\paren {-1}^n} {n^2} + i \dfrac {\cos n \theta} {n^2}\)

\(\ds \leadsto \ \ \)

\(\ds \sum_{n \mathop = 1}^\infty a_n\)

\(=\)

\(\ds \sum_{n \mathop = 1}^\infty \paren {\dfrac {\paren {-1}^n} {n^2} } + i \sum_{n \mathop = 1}^\infty \paren {\dfrac {\cos n \theta} {n^2} }\)

Both of the terms on the right hand side are convergent real series.

Hence from Convergence of Series of Complex Numbers by Real and Imaginary Part, $\ds \sum_{n \mathop = 1}^\infty a_n$ is convergent.

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.3$. Series: Example $\text{(i)}$