<s>[INST] determine whether this series converges for this value of z Does $$f(z)=\displaystyle \sum_{n=0}^{\infty} \frac{2^n+n^2}{3^n+n^3}z^n$$ converge for   $z=\frac{-3}{2}$?
 [/INST] $$ \frac{2^n + n^2}{3^n + n^3} z^n  = \frac{2^n}{3^n + n^3} z^n + \frac{n^2}{3^n + n^3} z^n $$
$$ \frac{n^2}{3^n + n^3} z^n < n^2 \left ( \frac{z}{3} \right )^n  = \frac{n^2 }{(-2)^n } \text{ Which converges from Ratio test }$$
$$ \frac{2^n}{3^n + n^3} z^n = \left ( \frac{2}{3} z\right )^n \frac{1}{1 + \frac{n^3}{3^n}}  \text{ which is } (-1)^n \text{ for } n \rightarrow \infty \text{ (It does not converge) } $$
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