Convergence of $g_k(x) = \Big{(} \frac{(-1)^k}{\sqrt{k}} \Big{)} \cos(kx)$

Question

From “Elementary Classical Analysis. (marsden, hoffman)

Let $g_k(x) = \Big{(} \frac{(-1)^k}{\sqrt{k}} \Big{)} \cos(kx)$ on $\mathbb{R}$.

Does the series $\sum g_k $ converge pointwise or uniformly.

Graphically, this seems to converge (at least pointwise) to something. I think I need a trig identity to get an upper bound for the series, I can't think of one.

Answer 1

The series does not converge when $x =\pi$ since $\cos (k \pi)=(-1)^{k}$ and $\sum \frac 1 {\sqrt k}$ is divergent.