Uniform convergence of that series

Question

Is $$ \sum_{n=1}^{\infty} \frac{(-1)^n} {1+nx} $$ uniformly convergent on $[\frac{1}{2}, 1]$? I tried using the M-test but it wasn't successful, I don't think I can use it for this series. Not sure how I could do it. Thank you.

Answer 1

After noticing that the series is alternating, we have the control

$$\left|\sum_{n = N}^{\infty} \frac{(-1)^n}{1 + nx}\right| \le \left|\frac{(-1)^N}{1 + Nx}\right| = \frac{1}{1 + Nx} \le \frac{1}{1 + N/2}$$

for all $x$ in the relevant range. Can you finish from here?

Answer 2

The series is uniformly convergent on any compact subset excluding $0$. This follows from Dirichlet's test.

The Dirichlet's test states that if $\sum f(x)$ has uniformly bounded partial sums and if $g_n(x)\to 0$ uniformly on some set $E \subset \mathbb{R}$, then $\sum f_n(x)g_n(x)$ converges uniformly on $E$. This can be proved by summation by part.