Existence of Interval of Convergence of Power Series/Corollary 2

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Theorem

Let $\displaystyle S \left({x}\right) = \sum_{n \mathop = 0}^\infty a_n x^n$ be a power series.

Let $S \left({x}\right)$ be convergent at $x = x_0$.


Then $S \left({x}\right)$ is convergent for all $x$ such that $\left\lvert{x}\right\rvert < \left\lvert{x_0}\right\rvert$.


Proof

Let $S \left({x}\right)$ converge when $x = x_0$.

Then by Existence of Interval of Convergence of Power Series, $x_0$ is in the interval of convergence of $S \left({x}\right)$ whose midpoint is $0$.

The result follows by definition of interval of convergence.

$\blacksquare$


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