Math 2.0

Sequences, series, summability Definition

Regular summability method

Builds on Summability methodConvergent sequence

Definition

Let M ⁣:DM→𝕂M \colon D_M \to \mathbb{K} be a summability method, where 𝕂\mathbb{K} is ℝ\R or ℂ\C. The method MM is regular if every convergent sequence ss belongs to DMD_M and M(s)=lim⁡n→∞sn.M(s) = \lim_{n \to \infty} s_n. Equivalently, every convergent series is MM-summable, and its MM-sum is its ordinary sum.

Remarks

Example: Cesàro summation is regular, because the Cesàro means of a convergent sequence converge to the same limit. Non-example: the method with DMD_M the convergent sequences and M(s)=2lim⁡snM(s) = 2 \lim s_n is a summability method, but it is not regular, since it sends the constant sequence 11 to 22. A method that sums every convergent sequence, but not necessarily to its ordinary limit, is called conservative.

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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 sumdefinitionSummability method, definition, no evidence levelSummability methoddefinitionRegular summability method, definition, no evidence levelRegular summability m…definition
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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. Summability method definition
  8. Regular summability method definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Divergent_series
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MSC40C99, 40D05
DefinesRegular summability method
UsesSummability method, Convergent sequence
Changed2 November 2026

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