Does the series diverge or converge: $\sum\limits_{n=1}^{\infty} \frac{1+(-1)^n}{\sqrt{n}}$? I'm asked to find out if this series converges or diverges:
$$\sum_{n=1}^{\infty} \frac{1+(-1)^n}{\sqrt{n}}=0+\frac{2}{\sqrt{2}}+0+\frac{2}{\sqrt{4}}...$$
So I thought I could use a direct comparison test, so
$$\sum_{n=1}^{\infty} \frac{1+(-1)^n}{\sqrt{n}}\leq \sum_{n=1}^{\infty}\frac{2}{\sqrt{n}}$$
But giving that this is a p-serie with $$p=-\frac{1}{2}$$ I know that I can not use this to compare with because it diverges. So I'm stuck. Does anyone have some tips?

Hint:
Since $1+(-1)^n=0$ if $n$ is odd and $2$ if $n$ is even, it's
$$\sum\limits_{k=1}^\infty\dfrac{2}{\sqrt{2k}}=
\sqrt2\sum\limits_{k=1}^\infty \dfrac1{\sqrt k}$$
