# Complex power series which converges absolutely on the boundary converges absolutely on a neighborhood of the boundary [closed]

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

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• Examples, examples... which examples did you try? – 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 $\forall|z|\leq1$.