# Suppose $\sum a_n x^n$ be a power series [closed]

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.

## closed as off-topic by user296602, Sahiba Arora, aes, Claude Leibovici, ahulpkeAug 11 '17 at 9:09

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• You can give a more informative title. – 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 $R\le1$.