When does $\sum_{n=0}^{\infty} \frac{2^n+n^2}{3^n+n^3}z^n$ converge? For what values of $z \in \mathbb{C}$ does the following series converge:
$$\displaystyle \sum_{n=0}^{\infty} \frac{2^n+n^2}{3^n+n^3}z^n\quad ?$$

You're given
$$f(z)=\displaystyle \sum_{n=0}^{\infty} \frac{2^n+n^2}{3^n+n^3}z^n$$
A sensible solution would be using Cauchy's Root test. We want to find
$$\lim\limits_{n\to\infty}\left(\frac{2^n+n^2}{3^n+n^3}\right)^{1/n} =$$
$$=\lim\limits_{n\to\infty}\frac 2 3\left(\frac{1+n^2/2^n}{1+n^3/3^n}\right)^{1/n} =$$
$$=\frac 2 3\left(\frac{1+0}{1+0}\right)^{0}=\frac 2 3 $$
Then the sum converges for $|z|<\dfrac 3 2 $
