Uniform Convergence of $\sum_1^\infty(-x)^n/n$ on $[a, 1]$

Question

How can I prove that $\sum_1^\infty(-x)^n/n$ converges uniformly on $[a, 1]$ where $a\in(-1,1)$

I've tried to use both Cauchy's criteria and Weierstrass M Test but have failed since the series does not converge at $-1$. I still think Cauchy criteria might work but I'm unable to find the right inquality I guess

In general, how can I prove that if a power series converges on $(-R,R]$ then in converges uniformly on $[a,R]$ where $a\in(-R,R)$

Answer 1

This follows from Dirichlet's test, since $\left(\frac1n\right)_{n\in\mathbb N}$ is monotonic and converges to $0$ and the partial sums of $\sum_{n=0}^\infty(-x)^n$ are uniformly bounded.

Answer 2

Apply Abel's Lemma to the sequences $(a_nR_n)$ and $(x/R)^n$ to get for all $n > m≥N$ and all $x\in[0,R]$ that $∣a_mR_m(x/R)^m+···+a_nR_n(x/R)^n∣\le \epsilon(x/R)^m < \epsilon$

Abel's Lemma States that if $b_1\ge b_2\ge\dots$ and $\sum_1^m a_n$ is bounded (for all $m$) (by say M) then $|a_nb_n+\cdots+a_mb_m|<Mb_0$