Convergence of a series involving $e^{in\phi}$

Question

Consider the following series:

$\displaystyle\sum_{n=1}^{\infty} \frac{(n+1)e^{in\phi}}{n^2}, \phi \in \mathbb{R}, \phi ≠ 2\pi k$ for $k \in \mathbb{Z}$

Then:

$\displaystyle\sum_{n=1}^{\infty} \frac{(n+1)e^{in\phi}}{n^2} = \displaystyle\sum_{n=1}^{\infty} \frac{e^{in\phi}}{n} + \frac{e^{in\phi}}{n^2} $

Form the respective lecture of my math course I know that

$\displaystyle\sum_{n=1}^{\infty} \frac{e^{in\phi}}{n}$ converges.

But I don't know how to make use of that nor do I know how to formulate the proof whether the series is convergent or even absolutely convergent.

Answer 1

That's the sum of two series ofr which the second one converges absolutely. And you know that the first one converges. So, your series converges.