TITLE: Is a power series uniformly convergent in its interval of convergence?
QUESTION [15 upvotes]: Let $R>0$ be the radius of convergence of a power series $Σa_nx^n$. Is it not uniformly convergent in $(-R,R)$? My book goes out of its way to say that if $[a,b]⊂(-R,R)$, then the power series converges uniformly in $[a,b]$. Can't we just say that it is uniformly convergent in $(-R,R)$?

REPLY [13 votes]: No. Think about the geometric series, which converges (but not uniformly) to 
$$
1 + x + x^2 + \cdots = \frac{1}{1-x}
$$
on $(-1,1)$.

REPLY [12 votes]: No. Example: $ \sum_{n=0}^{\infty}x^n=\frac{1}{1-x}$ for $|x|<1$.
Suppose that $s_n(x):=\sum_{k=0}^{n}x^k=\frac{1-x^{n+1}}{1-x}$ converges uniformly on $(-1,1)$ to $\frac{1}{1-x}$. 
Then , to $ \epsilon=1$ we get $N \in \mathbb N$ such that
$\frac{|x|^{N+1}}{1-x}=|s_N(x)-\frac{1}{1-x}|<1$  for all $x \in (-1,1)$.
Hence $\lim_{x \to 1-}\frac{|x|^{N+1}}{1-x} \le 1$. But $\lim_{x \to 1-}\frac{|x|^{N+1}}{1-x}= \infty$, a contradiction.