Let $$v_n=\frac{(-1)^n}{2\sqrt{n}+ cos(x)}$$
I am asked to prove that $\sum v_n$ converges uniformly.
This is my attempt: Let $$S_n(x)= \sum_{k=0}^{n}v_n$$ and $$F(x)=\sum_{k=0}^{\infty}v_n$$
I have to show that $|F(x)-S_n(x)|$ converges. But: $$|F(x)-S_n(x)|= \sum^{\infty}_{k=n+1}\frac{1}{2\sqrt{k}+cos(x)}$$
And this is where I am stuck. By looking at this sum, I would say that it diverges yet I am asked to prove it converges. Any help would be appreciated.