Divergent Series/Examples/sin i n over n^2

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Example of Divergent Series

The complex series defined as:

$\ds S = \sum_{n \mathop = 1}^\infty \dfrac {\sin i n} {n^2}$

is divergent.


Proof

\(\ds \cmod {\dfrac {\sin i n} {n^2} }\) \(=\) \(\ds \cmod {\dfrac {\exp \paren {i \paren {i n} } - \exp \paren {-i \paren {i n} } } {2 i n^2} }\) Sine Exponential Formulation
\(\ds \) \(=\) \(\ds \cmod {\dfrac {\exp \paren {- n} - \exp n} {2 n^2} }\)
\(\ds \) \(>\) \(\ds \dfrac {e^n - 1} {2 n^2}\)
\(\ds \) \(\to\) \(\ds \infty\)


Hence the result.

$\blacksquare$


Sources