Complex - Testing Convergence for $\sum_{n=1}^{\infty} n^{2}z^{n}$ Just wanting to make sure my conclusions are correct for this
$$\sum_{n=1}^{\infty} n^{2}z^{n}\quad \text{for}\quad |z| < 1 $$
When using the ratio and root test, i conclude that my series has either divergence or convergence. Totally not cool right?
So I then and use Dirichlet Test
https://en.wikipedia.org/wiki/Dirichlet%27s_test
Letting $a_n = n^{2}$ and $b_n = z^{2}$, its obvious that the function fails the first two tests. 
So will it then diverge?

The ratio test:
$$
   1 >  \lim_{n}\frac{(n+1)^2}{n^2}|z| =|z| .
$$
The root test:
$$
   1 > \lim_{n}(n^2|z|^{n})^{1/n} = \lim_{n}e^{2\frac{ln(n)}{n}}|z|=|z|.
$$
In either case, you have absolute convergence for $|z| < 1$ and divergence for $|z| > 1$.
