Check Uniform convergence of $\sum _{n=1}^{\infty }(\frac{\ln x}{x})^n$ in $[1,\infty)$

Question

Check Uniform convergence of $\sum _{n=1}^{\infty }(\frac{\ln x}{x})^n$ in $[1,\infty)$

I have hard time trying to check uniform convergence of $$\sum _{n=1}^{\infty }\left(\frac{\ln x}{x}\right)^n$$

one thing that I noticed that it is geometric series so I thought to use Weierstrass M-test but it didn't work with me

Answer 1

Note that $$ \left|\frac{\ln x}{x}\right |\leq \frac{1}{e}\,\,\text { for } x\in [1,\infty) $$ by an easy maximization problem. So use $M_n=\frac{1}{e^n}$ and conclude by the Weirstrass-M test.