I'm supposed to prove the Title.
I have tried to prove by contradiction and using the limit definition but with not much luck.
thanks for the help
2026-04-13 12:06:33.1776081993
if $\sum_{n=1}^{\infty} a_nx^n$ has a convergence radius $R<1$ then $\lim_{n\to\infty} a_n \neq 0$
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If $R<1$ , then $ \lim \sup |a_n|^{1/n}>1.$ Hence $|a_n|^{1/n}>1$ for infinitely many $n$, thus $|a_n|>1$ for infinitely many $n$. This gives the result.