I was recently reading that power series of form $\sum_{n=0}^\infty b_n(x-a)^n$ converge uniformly to some uniform limit function on compact intervals $[a-r,a+r]$ if $r$ is less than the radius of convergence.
I was curious about the case on an open, noncompact interval. Particularly, is there an example of a formal power series $\sum_{n=0}^\infty b_nx^n$ which is pointwise convergent on $(-1,1)$ but does not converge uniformly?
Best Answer
The series $\sum_{n=0}^\infty x^n$ converges point-wise on the interval $(-1,1)$ to the function $\frac{1}{1-x}$. If it were to converge uniformly on $(-1,1)$, then the function would have to be bounded, but it is not. So, this is an example as you are looking for.