Math 2.0

definition

Absolute convergence

Definition

Let (an)(a_n) be a real sequence. The series ∑an\sum a_n converges absolutely if the series ∑∣an∣\sum |a_n| of absolute values converges.

Remarks

The series ∑∣an∣\sum |a_n| has nonnegative terms. So it converges exactly when its partial sums are bounded. Absolutely convergent series converge, but the converse fails. The alternating harmonic series converges, but not absolutely. A convergent series that does not converge absolutely is conditionally convergent. The same definition works for complex terms, with the modulus in place of the absolute value.

Explanations

L1 human

A series converges absolutely if it still converges after we make all its terms nonnegative. Then cancellation between positive and negative terms is not needed for convergence.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionPartial sum, definition, no evidence levelPartial sumdefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionAbsolute convergence, definition, no evidence levelAbsolute convergencedefinition
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  1. Sequence definition
  2. Real sequence definition
  3. Limit of a sequence definition
  4. Series definition
  5. Convergent sequence definition
  6. Partial sum definition
  7. Convergent series definition
  8. Absolute convergence definition

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Sources

  1. Web https://en.wikipedia.org/wiki/Absolute_convergence, Definition for real and complex numbers

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