I'm learning about uniform convergence, and I wanted to ask if I'm properly using Cauchy's test for uniform convergence to prove the series $\sum_{n=1}^{\infty} (nx-n+1)x^n$ doesn't uniformly converge in $(-1,1)$ (I already know its 'convergence range' is $(-1,1)$):
The test gives us that, assuming there's uniform convergence, for every $l > 0$ we have an $N$ such that if $m,n>N$, $|\sum_{k=n}^{m} f_k(x)|
Am I properly using the test? Is there an even simpler way to do do this (maybe using a different test)? Thanks!