# Does this series converge uniformly?

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

Use Dirichlet's test. Adapt the proof in the link to show that $|S_n(x) - S_m(x)| \to 0$ as $m,n \to \infty$ uniformly in $x$. For this you have to show that the partial sums $\sum_{k=1}^N (-x)^k$ are bounded uniformly in $N$ ${\it and}$ $x\in [0,1]$.