Divergent Series/Examples/(n + i) over n^2
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Example of Divergent Series
The complex series defined as:
- $\ds S = \sum_{n \mathop = 1}^\infty \dfrac {n + i} {n^2}$
is divergent.
Proof
\(\ds S\) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \dfrac {n + i} {n^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \dfrac n {n^2} + i \sum_{n \mathop = 1}^\infty \dfrac 1 {n^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \dfrac 1 n + i \sum_{n \mathop = 1}^\infty \dfrac 1 {n^2}\) |
From Harmonic Series is Divergent, $\ds \sum_{n \mathop = 1}^\infty \dfrac 1 n$ is a divergent series.
The result follows from Convergence of Series of Complex Numbers by Real and Imaginary Part.
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.3$. Series