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.


\(\ds S\) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \dfrac {n + i} {n^2}\)
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \dfrac n {n^2} + i \sum_{n \mathop = 1}^\infty \dfrac 1 {n^2}\)
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \dfrac 1 n + i \sum_{n \mathop = 1}^\infty \dfrac 1 {n^2}\)

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.