If a complex power series $\sum_{n = 0}^{\infty} a_n z^n$ converges absolutely for $|z| \leq 1$, does it necessarily converge absolutely for $|z| < 1 + \epsilon$, for some $\epsilon > 0$?


closed as off-topic by Did, Shaun, Daniel W. Farlow, pjs36, quid Apr 28 '15 at 21:52

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Shaun, pjs36, quid
If this question can be reworded to fit the rules in the help center, please edit the question.

  • 1
    $\begingroup$ Examples, examples... which examples did you try? $\endgroup$ – Did Apr 18 '15 at 20:38

It's false. For instance, $\sum_{n=1}^{+\infty}\frac{z^n}{n^2}$ has ray of convergence $1$, but it does converge absolutely $\forall|z|\leq1$.


Not the answer you're looking for? Browse other questions tagged or ask your own question.