I'm currently working through a homework problem, but I've been stumped on this one problem for almost a day now. I don't want the problem to be worked out, rather I just want some idea of where to start/if my thought process has been incorrect.
The problem is to figure out whether or not the following series satisfies the M-test and converges uniformly on the interval $[0,1]$.
$$\sum_{k=1}^\infty \frac{e^{x/k}-1}{k}$$
I initially tried to see if $$\sum_{k=1}^\infty \frac{e^{1/k}-1}{k}$$ converges, since $$\frac{e^{x/k}-1}{k} \le \frac{e^{1/k}-1}{k}$$ when $x \in [0,1]$. However, after trying most of the convergence tests (Integral, comparison, ratio, root, etc) I wasn't able to conclude if this bound converges or not.
However, I've also been unable to show if $\sum_{k=1}^\infty \frac{e^{x/k}-1}{k}$ diverges, which would mean that it would fail the M-test.
Does anybody have any suggestions as to where to start with this problem?
$$\sum_{k=1}^\infty\frac{e^{1/k}-1}k=\sum_{k=1}^\infty\sum_{j=1}^\infty\frac1{k^{j+1}j!}$$ $$=\sum_{j=1}^\infty\frac1{j!}\sum_{k=1}^\infty\frac1{k^{j+1}}=\sum_{j=1}^\infty\frac{\zeta(j+1)}{j!}$$ $$<\sum_{j=1}^\infty\frac2{j!}=2(e-1)$$ Therefore the bound converges and the first series converges uniformly.