Limit of $a_n$ is -$2$ then consider $\sum a_n ^{-n}$

Question

If the limit of $a_n$ is -$2$ then consider the infinite series $\sum a_n ^{-n}$. Does this converge or diverge?

I thought that because of the exponent, the root test will be a good approach. Consider:

$$ \lim_{n\rightarrow \infty} \sqrt[n]{a_n^{-n}}=\lim_{n\rightarrow \infty} \frac{1}{a_n}= -\frac{1}{2}<1$$ so the limit converges. Am I correct in saying this?

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Edit: a condittion for the root test is that $a_n>0$ this is clearly not the case, my proof fails.

Answer 1

We have that

$$a_n\sim \left(-\frac{1}{2}\right)^n$$

which clearly converges by geometric series.

As an alternative consider the $\sum |a_n|$ and apply root test if you want use that.