TITLE: Is there a real power series with radius of convergence 1 that converges uniformly on (−1,1)?
QUESTION [5 upvotes]: Is there a real power series with radius of convergence $1$ that converges uniformly on $(−1,1)$?

I am guessing the answer is yes, if we can construct a function with power series such that it converges on the radius of convergence, but I do not know any examples of this. Help someone?

REPLY [5 votes]: Yes take $$\sum _ {n=1}^{\infty}\frac{x^n}{n^2}$$
The uniform part can be checked with M-test Weiestrass. And the radius with $\limsup \sqrt[n]{ n^2 }=1$