Do these series converge uniformly?

Question

1. $$\sum_{n \geq 0}e^{-nx}\cos(nx),\quad x \in \mathbb R$$

2. $$\sum_n (-1)^n \frac {x^2+n}{n^2},\quad x\in \mathbb R$$

I tried using Weierstrass M-test but that does not work here. I think these series don't converge uniformly but I can't prove it.

Answer 1

1. The series of functions doesn't even converge pointwise on all $\mathbb R$. You'll notice that for $x =-2 \pi$: $$e^{-nx} \cos nx = e^{2 n \pi}$$ diverges.

2. A series of functions $\sum f_n(x)$ that is uniformly convergent is Cauchy uniformly convergent. This implies that $\sup \vert f_n(x) \vert \to 0$ as $ n \to \infty$. This is not the case for this series of functions as for $x=n$: $$\left\vert (-1)^n \frac{x^2 +n}{n^2} \right\vert = \frac{n^2+n}{n^2} > 1.$$