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?

Other way to see it is that in the sum you take common factor of $\sqrt2$ u get $$\frac{1}{\sqrt2}\left(\frac{2}{\sqrt1}+\frac{2}{\sqrt2}+\frac{2}{\sqrt3}+...\right)=\frac{1}{\sqrt2}\sum_{n\geq1} \frac{2}{\sqrt n}=\frac{2}{\sqrt2}\sum_{n\geq1} \frac{1}{\sqrt n}>\infty$$
