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Let $\sum\limits_{n=0}^{+\infty}a_nx^n$ be a power series. Prove that

  1. If $\large \lim\limits_{n\to\infty}a_ns^n=0,s>0$, then the power series above converges absolutely for $|x|<s$.
  2. Conclude that
    $$R=\sup A,$$ where $R$ is the radius of convergence of the power series and $$A=\left\{x\geq 0|\lim_{n\to+\infty}a_nx^n=0\right\}$$

The (2) item follows straightforward from the (1) item, but here is the trouble, how can we prove (1)?

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  • $\begingroup$ Maybe compare the power series for points with $|x|<s$ with the given series? $\endgroup$ – Karl Sep 21 '14 at 12:41
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Let $x$ be such that $|x|\lt t$. Notice that for $n$ large enough, we have $$|a_nx^n|\leqslant |a_n t^n|\left|\frac xt\right|^n\leqslant \left|\frac xt\right|^n,$$ and the series $\sum_nr^n$ converges for $r\in (0,1)$.

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