Suppose $\sum a_n x^n $ be a power series. Prove that if $0<\lim\sup|a_n|<\infty$, then the power series has radius of convergence $R=1$.

I really have no idea where I have to start can anyone help me with the problem please? Thanks for your help in advance.


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    $\begingroup$ You can give a more informative title. $\endgroup$ – Ranc Aug 3 '17 at 6:29

This is not true. All you can deduce is that $R\ge1$. This is straightforward: the condition implies that $\sum_n a_nx^n$ converges for $|x|\le1$, by comparison.


This is textbook stuff. The sum converges for $|x|<1$ by comparison with the geometric series, so $R\ge1$. But it diverges for $x=1$ as $|a_n| \not\to0$, so $R\le1$.

  • $\begingroup$ ...sorry i edited question now $\endgroup$ – user293581 Aug 3 '17 at 4:26
  • $\begingroup$ ...i am really sorry i had mistake with writing question can please answer me now thank you $\endgroup$ – user293581 Aug 3 '17 at 4:27
  • $\begingroup$ @Lord Shark the Unknown ..but question says radis of converence =1 $\endgroup$ – Inverse Problem Aug 3 '17 at 4:35