consider the series $\sum_{n=1}^{\infty}\frac{(-x)^n}{n},$ with $x\in[0,1].$ I am wondering if this series converge uniformly or not? I could prove that for all $[0,\alpha]$ with $\alpha<1$ the series is uniformly convergent to $\ln(1+x)$. Therefore if the series is uniformly convergent its limit should be $\ln(1+x)$. But I have no idea how to prove or disprove whether the series converges uniformly... Does anyone have an idea?
Best wishes