Math 2.0

Functions of a complex variable Definition

Interval of convergence

Builds on Power seriesRadius of convergence

Definition

Let ∑an(x−c)n\sum a_n (x - c)^n be a real power series. Its interval of convergence is the set of real numbers xx for which the series converges.

Remarks

Let RR be the radius of convergence. The interval of convergence contains (c−R,c+R)(c - R, c + R) and lies in [c−R,c+R][c - R, c + R]. Each endpoint must be checked on its own. For ∑xn/n\sum x^n / n, the interval is [−1,1)[-1, 1). For ∑xn\sum x^n it is (−1,1)(-1, 1), and for ∑xn/n2\sum x^n / n^2 it is [−1,1][-1, 1].

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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 convergencedefinitionInterval of convergence, definition, no evidence levelInterval of convergen…definition
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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
  7. Interval of convergence definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Radius_of_convergence, Convergence on the boundary
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MSC30B10
DefinesInterval of convergence
UsesPower series, Radius of convergence

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