about Dirichlet test in convergence

Question

whether this infinite sequence $\sum_{n=1}^\infty \frac{(-1)^ncos(nx)}{\sqrt n}$ converge on $\mathbb R$. I know that it would converge uniformly intuitivly on [$-\pi+\delta$, $\pi-\delta$] where $\delta$ is from $-\pi$ to $\pi$. However, I don't know how to prove my intuition. Thanks for helping.

Answer 1

Note that, for any $N\in\mathbb N$, and $x$ in such an interval,\begin{align}\left\lvert\sum_{n=1}^N(-1)^n\cos(nx)\right\rvert&=\left\lvert\sum_{n=1}^N(-1)^n\operatorname{Re}(e^{inx})\right\rvert\\&=\left\lvert\operatorname{Re}\left(\sum_{n=1}^N(-e^{ix})^n\right)\right\rvert\\&=\left\lvert\operatorname{Re}\left(\frac{-e^{ix}-(-1)^{N+1}e^{(N+1)ix}}{1+e^{ix}}\right)\right\rvert\\&\leqslant\left\lvert\frac{-e^{ix}-(-1)^{N+1}e^{(N+1)ix}}{1+e^{ix}}\right\rvert\\&\leqslant\frac2{\bigl\lvert1+e^{ix}\bigr\rvert}.\end{align}Can you take it from here?