Radius of Convergence
Every power series has an : absolute convergence strictly within distance of the centre, divergence beyond, nothing enforced on the boundary.
Let be a power series. Then is called the radius of convergence of the power series, if
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converges absolutely for all such that .
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if , then diverges for all such that .
Existence
Convergence within the radius
Proof
Let , which contains . If is unbounded, set : absolute convergence then holds on all of by the comparison below. Otherwise set . For the series diverges by definition of .
If , choose with , and with . Convergence of makes its terms bounded, say , so
Comparison with a geometric series shows that converges absolutely. In particular every power series converges absolutely inside its radius of convergence and diverges outside it.
Related
Stated in
- Definition 5.7 (Radius of Convergence)§5.2 Basics on Power Series
- Proposition 5.8§5.2 Basics on Power Series
- Lemma 5.6§5.2 Basics on Power Series