Weierstrass M-test help

Question

I am supposed to use M-test on this one $$\sum \frac {n\ln (1+nx)}{x^n}$$ on $$1<x< \infty$$ But I face problems finding an appropriate $M_n$, thanks for help

Answer 1

Note that for $x \geq r > 1$,

$$\frac {n\ln (1+nx)}{x^n}< \frac {n(nx)}{x^n}= \frac {n^2}{x^{n-1}} \leqslant \frac {n^2}{r^{n-1}}.$$

We have

$$\lim_{n \to \infty}\left[\frac {n^2}{r^{n-1}}\right]^{1/n}=\lim_{n \to \infty}\frac {(n^{1/n})^2r^{1/n}}{r}=\frac1{r} < 1.$$

By the root test, the series $\sum n^2/r^{n-1}$ converges and

$$\sum\frac {n\ln (1+nx)}{x^n}$$

converges uniformly on $[r,\infty)$ by the M-test for any $r > 1$.