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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
HINT:
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 $N$ ${\it and}$ $x\in [0,1]$.
