Answer
Converges absolutely.
Work Step by Step
Consider $a_n= \dfrac{(-2)^{n+1}}{n+5^n}$
$|a_n|=\dfrac{2^{n+1}}{n+5^n}=(2) \dfrac{2^{n}}{n+5^n}$
we see that $ \Sigma_{n=1}^\infty (2)\dfrac{2^{n}}{n+5^n} \leq \Sigma_{n=1}^\infty (2) \dfrac{2^n}{5^n}
=2 \Sigma_{n=1}^\infty (\dfrac{2}{5})^n$ (a convergent p-series)
Thus, the series Converges absolutely.
