Existence of Interval of Convergence of Power Series/Corollary 2
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Theorem
Let $\ds \map S x = \sum_{n \mathop = 0}^\infty a_n x^n$ be a power series.
Let $\map S x$ be convergent at $x = x_0$.
Then $\map S x$ is convergent for all $x$ such that $\sequence x < \sequence {x_0}$.
Proof
Let $\map S x$ converge when $x = x_0$.
Then by Existence of Interval of Convergence of Power Series, $x_0$ is in the interval of convergence of $\map S x$ whose midpoint is $0$.
The result follows by definition of interval of convergence.
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.9$: Generating Functions