Uniform convergence of $\sum_{n=1}^{\infty}(\sum_{k=0}^{m}kn^k)x^n$ $m\in \mathbb{N}$ Some constant

Question

How can I prove or disprove uniform convergence $\sum_{n=1}^{\infty}(\sum_{k=0}^{m}kn^k)x^n$ $m\in \mathbb{N}$ Some constant

Answer 1

The quantity $a_n:=\sum_{k=1}^mkn^k $ can be evaluated explicitly, but it suffices to estimate it as follows $$ m\,n^m\le a_n\le m^2\, n^m$$ The radius of convergence of the series $\sum_{n=1}^\infty a_nx^n$ is equal $1.$ The series is not convergent at $x=\pm 1.$ Thus uniform convergence holds for $|x|\le r$ and any fixed $r<1.$