Taylor polynomial of a complex function

Find Taylor's polynomial of the following function in $\displaystyle z_0=-1$ : $\displaystyle f(z)=\frac{1}{z^2}$.

Determine the radius of convergence of the series.

My attempt :

$\displaystyle f(-1)=\sum _{n=0}^{\infty} \frac{(-1)^n(n+1)}{z^{n+2}}(z+1)^n$. Which answers the first question.

Now trying to figure out the second question :

So $\displaystyle a_n=\frac{(-1)^n(n+1)}{z^{n+2}}$.

I apply the ratio test : $\displaystyle \left| \frac{a_{n+1}}{a_n} \right| =\frac{(-1)(n+2)}{z}$ which tends to $\displaystyle -\frac{1}{z}$ when $\displaystyle n$ tends to $\displaystyle \infty$.

Hence the radius of convergence is ...? It makes no sense at all.

I'd like to know where is(are) my error(s). Thanks a lot!