Prove: $\sum_{n=1}^{\infty}\frac{-\ln(n)}{n^x}$, $x\in(1,\infty)$ converges uniformly

Question

Prove: $\sum_{n=1}^{\infty}\frac{-\ln(n)}{n^x}$, $x\in(1,\infty)$ converges uniformly

My attempt:

For every $x>2$

$$\left|\frac{-\ln(n)}{n^x}\right|=\frac{\ln(n)}{n^x}\le\frac{n}{n^x}=\frac{1}{n^{x-1}}\le \frac{1}{n^{1+\epsilon}}.$$

Therefore converges uniformly by the M-test. But I'm having trubles proving for $1<x<2$

Answer 1

Hint: If each $f_n$ is bounded on a set $E$ and $\sum f_n$ converges uniformly to $f$ on $E,$ then $f$ is bounded on $E.$ In your problem, each summand is bounded on $(1,\infty).$ Is the sum bounded on $(1,\infty)?$