Find if $\sum_{n=1}^{\infty} (nx-n+1)x^n$ converges uniformly for $|x| < 1$.

Question

Find if $$\sum_{n=1}^{\infty} (nx-n+1)x^n$$
converges uniformly for $|x| < 1$.

I can tell that the partial sum is $s_n(x) =nx^{n+1}$, thus the sequence converges to 0. What can I do from here?

Answer 1

You're correct in that $s_n(x)=nx^{n+1}$, and for $\vert x \vert < 1$, this does converge to $0$. However, $s_n\left(1-\frac{1}{n+1}\right)=n\left(1-\frac{1}{n+1}\right)^{n+1}\sim \frac{n}{e}$, which does not tend to $0$.

Note that a sequence of functions $f_n\to f$ uniformly on $S$ if there exists a sequence $r_n\to0$ such that $\vert f_n(x) - f(x)\vert\le r_n$ for all $n$, $x\in S$. This counterexample shows that no such sequence can exist for the given functions.