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!

Let $X$ be a set. If $(f_n)$ is a sequence of functions from $X$ to $\mathbb C$, and if the series $\sum_{n\ge0}f_n$ converges uniformly on $X$, then the sequence $(f_n)$ converges uniformly to $0$.
Proof: Write
$$
f_n=\sum_{k=0}^n\ f_k-\sum_{k=0}^{n-1}\ f_k
$$
and let $n$ tend to $\infty$.
