For $\sum_{0}^{\infty}a_nx^n$ with an infinite radius of convergence, $a_n \neq0$ ,  the series does not converge uniformly  in $(-\infty , \infty)$. I'd like your help with the following question:

Let $\sum_{0}^{\infty}a_nx^n$ be a power series with an infinite
  radius of convergence, such that $a_n \neq 0 $ , for infinitely many values of
  $n$. I need to show that the series does not converge uniformly   in
  $(-\infty , \infty)$.

I quite don't understand the question, since I know that within the radius of convergence for a power series, there's   uniform convergence, and since I know that the radius is "infinite", it says that the uniform convergence is infinite, no?
Thanks!

Hint: put $s_N(x):=\sum_{n=0}^Na_nx^n$. If the sequence $\{s_N\}$ is uniformly convergent on $\mathbb R$ then the sequence $\{s_{N+1}-s_N\}$ converges uniformly on $\mathbb R$ to $0$ so $\{a_{N+1}x^{N+1}\}$ converges uniformly to $0$. Do you see the contradiction?
