Convergent Complex Series/Examples/1 over n^2 - i n
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Example of Convergent Complex Series
The series $\ds \sum_{n \mathop = 1}^\infty a_n$, where:
- $a_n = \dfrac 1 {n^2 - i n}$
is convergent.
Proof
\(\ds \sum_{n \mathop = 1}^\infty \cmod {\dfrac 1 {n^2 - i n} }\) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \cmod {\dfrac {n^2 + i n} {\paren {n^2 - i n} \paren {n^2 + i n} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \cmod {\dfrac {n^2 + i n} {n^4 + n^2} }\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \sum_{n \mathop = 1}^\infty \cmod {\dfrac {n^2 + i n} {n^4} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \cmod {\dfrac {n + i} {n^3} }\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \sum_{n \mathop = 1}^\infty \dfrac 2 {n^2}\) |
Thus $\ds \sum_{n \mathop = 1}^\infty \dfrac 1 {n^2 - i n}$ is absolutely convergent.
The result follows from Absolutely Convergent Series is Convergent: Complex Numbers.
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4$. Elementary Functions of a Complex Variable: Exercise $2 \ \text {(i)}$