Prove $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}x^k$ converges uniformly on $[0,1]$.
I don't think this series is uniformly convergent on $[0,1]$. If I let $x=0$, then $s_n=\sum\limits_{k=1}^{n}\frac{(-1)^{k+1}}{k}(0)^k \to 0$ as $n\to\infty$. If I let $x=1,$ then $s_n=\sum\limits_{k=1}^{n}\frac{(-1)^{k+1}}{k} \to \ln2$ as $n\to\infty$. So, the series is not uniformly convergent since for different values of $x$, I get different limits.
Is my reasoning valid? If not, how can you prove the series is uniformly convergent using the alternating series test?