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

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.

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 $$z=x+iy$$.