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

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.

$\blacksquare$