Math 2.0

definition

Radius of convergence

Definition

Let ∑an(x−c)n\sum a_n (x - c)^n be a real power series. Its radius of convergence is R=sup⁡{r≥0:the sequence (∣an∣rn) is bounded},R = \sup \{ r \ge 0 : \text{the sequence } (|a_n| r^n) \text{ is bounded} \}, an element of [0,∞][0, \infty].

Remarks

The series converges absolutely for ∣x−c∣<R|x - c| < R, and it diverges for ∣x−c∣>R|x - c| > R. At the points x=c±Rx = c \pm R, the series may converge or diverge. Many texts define RR as the supremum of all ∣x−c∣|x - c| where the series converges. The two definitions give the same value. The Cauchy–Hadamard theorem gives a formula for RR. Mathlib uses the bounded form, in FormalMultilinearSeries.radius.

Explanations

L1 human

A power series converges on an interval centered at cc, and diverges outside it. The radius of convergence is the half length of this interval.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionPower series, definition, no evidence levelPower seriesdefinitionRadius of convergence, definition, no evidence levelRadius of convergencedefinition
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  1. Sequence definition
  2. Real sequence definition
  3. Bounded sequence definition
  4. Series definition
  5. Power series definition
  6. Radius of convergence definition

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Sources

  1. Web https://en.wikipedia.org/wiki/Radius_of_convergence, Definition

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