I have this series of functions $\sum_{n=1}^\infty (-1)^n\log(1+\frac{x}n)$ with $x\geq0$. It's easy to see the pointwise convergence of the series and I also prove that it converges uniform on compact set. The problem is when $x$ is going to infinity: I conjecture that the convergence is not uniform in the whole interval $[0,\infty)$ but I have some problem to prove it.
Do you have any idea?