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I have a somewhat theoretical question to the definition of the radius radius of convergence of infinite power series. According to the definition for a power series $\sum_{n=0}^\infty a_nx^n$ radius of convergence $\rho \ge0$. For $|x|<\rho$ the series converges at $-\rho<x<\rho \\$.

Do I understand it correctly, that if $\rho=0$, the series won't converge?

I browsed several articles on Wikipedia, but didn't find an answer.

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  • $\begingroup$ If $\rho=0$, the series converges only when $x=0$. $\endgroup$ – David Mitra Jun 12 '14 at 16:24
  • $\begingroup$ @DavidMitra thanks, now it seems pretty straightforward $\endgroup$ – Dmitry Kazakov Jun 12 '14 at 16:25
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No, you don't.

The sum always converges absolutely for $-\rho < x < \rho$, and may or may not converge when $|x|=\rho$. Clearly if $\rho=0$, the series converges when $x=0$ (and only then).

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  • $\begingroup$ Thanks, that what I missed, it may or may not converge in case $|x|=\rho$. And I got the zero case as well. $\endgroup$ – Dmitry Kazakov Jun 12 '14 at 16:27

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