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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?

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  • $\begingroup$ Of course the concept mentioned in the title, "absolute convergence," is different from the concept in the question, "uniform convergence." $\endgroup$
    – Jesse Madnick
    Commented Jun 24, 2013 at 8:51
  • $\begingroup$ @JesseMadnick Thanks, I miswrote that. $\endgroup$
    – Nastassja
    Commented Jun 24, 2013 at 8:54
  • $\begingroup$ In fact, the convergence is uniform if and only if it converges at both end points $\endgroup$
    – Anshuman Agrawal
    Commented Mar 20, 2023 at 15:51

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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.

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