Provide a series that converges uniformly, but for which we can't apply Weierstrass M-test

Question

Provide a series that converges uniformly, but for which we can't apply Weierstrass M-test. My guess is an exponential: $$e^x=\sum\limits_{n=0}^{\infty}\frac{x^n}{n!}$$ on the interval $x\in(1;+\infty)$, because we can't bound $x^n$ on that interval. For example: I can choose $x=(n!)^n$. That means that my $f_n(x)=(n!)^{n-1}$ which can't be bounded by such $M_n$ that the series of it converges

Answer 1

You can take, for instance,$$\displaystyle\sum_{n=1}^\infty\frac{(-1)^n}nx^n\quad (x\in[0,1]).$$It converges uniformly (by Abel's theorem), but you cannot apply the Weierstrass $M$-test. If you could, it would also converge absolutely for each $x\in[0,1]$. But it only converges conditionally when $x=1$.