Answer
Converges Absolutely
Work Step by Step
Let us consider $p_n=(-1)^{n+1} \dfrac{2^{n+1}}{n+5^n}$ and $p_n$ refers to the all positive values of $n$.
Also, $|p_n|=\dfrac{2^{n+1}}{n+5^n}=(2) \dfrac{2^{n}}{n+5^n}$
Apply the limit comparison Test.
$\implies \Sigma_{n=1}^\infty (2)\dfrac{2^{n}}{n+5^n} \leq \Sigma_{n=1}^\infty (2) \dfrac{2^n}{5^n}
=(2) \Sigma_{n=1}^\infty (\dfrac{2}{5})^n$
But the series $\Sigma_{n=1}^\infty (\dfrac{2}{5})^n$ shows a convergent $p$-series.
Hence, the series Converges Absolutely.