Uniform convergence of series of functions given convergence of coefficients

Question

Prove that if $\sum{a_n}$ converges then the series $\sum{a_nx^n}$ converges uniformly on [0,1]

I believe that I must use the Weierstrass M-test to show this convergence, but this requires that the series be non-negative. I'm not sure where to go from here

Answer 1

One form of Abel's theorem states that if a power series converges at end-point $R$ of its interval of convergence, then it converges uniformly on $[0,R]$.

But if this was supposed to be an easy exercise in elementary real analysis, I suspect the intention was to have $\sum_n a_n$ converge absolutely, or maybe $a_n \ge 0$.