$\sum_{1}^{\infty} \frac{1}{(1+nz)^k}$ is absolutely convergent.

Question

I'm trying to show that the series $$\sum_{1}^{\infty} \frac{1}{(1+nz)^k}$$ is absolutely convergent, with $z \in \mathbb{C}- \mathbb{R}$ and $k \in \mathbb{Z}$ such that $k > 1$, but I'm clueless in how to proceed. The problem itself doesn't seems to hard, but I do not know where to start, as I am very new to complex analysis.

In this case, I'm primarily interested in hints, but any help would be very welcomed.

Answer 1

Hint: Compare with a p-series. That is, the series $\sum_n1/n^p$ converges for $p\gt1$.

Also, note $\mid 1+nz\mid\ge \mid y\mid n$, where $z=x+iy$.