I'd like your help with proving that following sum uniformly converges:
$\sum_{n=1}^{\infty}\left(\left(n+\frac{x}{n}\right)\ln\left(1+\frac{x}{n}\right)-x\right)$ for $x \in [0,a]$.
I tried to use the theorem saying that if there's one point (in our case $x=0$), which the series pointwise converges, and the sum of U'_n converges also, the original sum uniformly converges, but it didn't work for me here.
Any hints?
Thanks a lot!