Complex - Testing Convergence for $\sum_{n=1}^{\infty} n^{2}z^{n}$

Question

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?

Answer 1

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 $|z| > 1$.