Is the series uniformly convergent?

Question

Let $$f(x)=\sum_1^\infty (-1)^{n+1}\frac{e^{-n^2x^4}}{n}.$$ Is $f$ uniformly convergent on [0,1] and in $\mathbb{R}_+$? I tried using comparison test but failed.

Answer 1

Hint. Let $a_n(x)=\frac{e^{-n^2x^4}}{n}$, then for $x\in \mathbb{R}$, $a_n(x)$ goes to $0^+$ monotonically as $n\to \infty$. Therefore, for $N\geq 1$, $$\left|\sum_{n=1}^\infty (-1)^{n+1}a_n(x)-\sum_{n=1}^N (-1)^{n+1}a_n(x)\right|\leq a_{N+1}(x).$$