Convergence of $\sum\limits_{n=1}^{\infty} {\dfrac{(-1)^{n-1}} {(\sqrt{n}+(-1)^{n-1})^p}}$

Question

Find $p$ that makes $\sum\limits_{n=1}^{\infty} {\dfrac{(-1)^{n-1}}{(\sqrt{n}+(-1)^{n-1})^p}}$ converge. Which $p$ makes the series converge absolutely?

I think that it converges for $p>0$, can I use: ${\dfrac{1}{{{{\left( {\sqrt{n} + {{\left( { - 1} \right)}^{n - 1}}} \right)}^p}}}} \sim \dfrac{1}{{{n^{\frac{p}{2}}}}}$ to conclude the series converges absolutely for $p>2$?

Thanks in advance!