How can we choose $z$ to make this series absolute convergent? I have the following series:
$\sum_{n=0}^\infty \frac{n^{\ln (n)}}{3^n} (z+2)^n$
The question is, how can we choose $z \in \mathbb{C}$ to make this absolute convergent?
For the first sight it seemed impossible to have such $z$, since $n^{ln n}$ should be way way bigger than $3^n$ as $n \rightarrow \infty$. Am I wrong? Any hints, ideas?
Thanks!

To estimate $n^{\ln n}$, you can use
$$n^{\ln n} = e^{\ln(n^{\ln n})} = e^{\ln n \ln n} = e^{\ln^2 n}$$
and
$$3^n = e^{\ln(3^n)} = e^{n \ln 3}$$
Which should tell you which of these grows quicker (does $n$ grow quicker than $\ln^2 n$?)
