Show $\sum_\limits{n=1}^{\infty}(1+\frac{1}{n})^{n^2}z^n$ is convergent.

Question

Show that the following series converge on $\mathbb{C}$:

$\sum_\limits{n=1}^{\infty}(1+\frac{1}{n})^{n^2}z^n$ where $z\in\mathbb{C}$ is a variable.

I applied the Abel Dirichlet´s test:

If I choose $|z|<1$ then $\lim_{n\to\infty}z^n=0$

However my problem lies with $\sum_\limits{n=1}^{\infty}(1+\frac{1}{n})^{n^2}$ that I am trying to prove its sum can be majored. I have tried several tests (root test,ratio test,Weierstrass test). All seem to fail. I noticed that $\lim_{n\to\infty}(1+\frac{1}{n})^{n^2}=e^2>1$

However I am supposed to prove $\sum_\limits{n=1}^{\infty}(1+\frac{1}{n})^{n^2}z^n$ is convergent.

Question:

How should I prove it? What am I doing wrong?

Answer 1

Using the root test: $$ \lim_{n\to\infty}\root n\of{|(1 + 1/n)^{n^2}z^n|} = \lim_{n\to\infty}{(1 + 1/n)^n}|z| = e|z| $$ and the series is convergent for $|z| < 1/e$.