# Is there a real power series with radius of convergence 1 that converges uniformly on (−1,1)?

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?

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Hint: Pick any power series which converge on $[-1,1]$. – N. S. Apr 9 '13 at 23:29

## 1 Answer

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 $\limsup \sqrt[n]{ n^2 }=1$

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bravo, thanks a bunch! – Lost1 Apr 9 '13 at 23:28
@Lost1 Glad I could help:) – clark Apr 9 '13 at 23:40