Investigating whether the series converges uniformly using the Weierstrass M-test.

Question

I have recently found myself stuck on a question for a long time, unable to get my head around it. It is:

I have to investigate whether the following series converges uniformly

$\sum_{n=1}^\infty\frac{(lnx)^n}{(x-1)^{2n}}$ on the interval $(3,\infty]$.

I have used the fact that $lnx \le x - 1$ for all $x \ge 1$ to show that $(lnx)^n \le (x-1)^n$ for all $x \ge 1$ but I do not know where to proceed from here. Thanks.

Answer 1

Note that $$ \frac{(\ln x)^n}{(x - 1)^{2n}} \leq \frac{(x - 1)^n}{(x - 1)^{2n}} = \frac{1}{(x - 1)^{n}} \leq \frac{1}{2^n} $$ by our assumption that $x \geq 3$. Now use the $M$ test with $f_n(x) = \frac{(\ln x)^n}{(x - 1)^{2n}}$ and $M_n = \frac{1}{2^n}$.