## $\sum_{n\in \mathbb{Z}} n^m e^{i n x }$

This is an exercise in the book ‘integral transforms and their applications’ by Davies.

The problem is to evaluate the series as a generalized function.

$$\sum_{n\in \mathbb{Z}} n^m e^{i n x } ,$$

with $$m > 0$$.

If $$m =0$$, I can recognize it as a series of delta functions. But I cannot envision what the function is like for $$m> 0$$.

How to proceed?