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?