Math 2.0

Numerical analysis Definition

Series acceleration

Builds on Limit of a sequencePartial sumSeries

Definition

A sequence transformation TT assigns to a sequence (sn)(s_n) of real or complex numbers a new sequence (tn)=T(sn)(t_n) = T(s_n). Each tnt_n depends on finitely many terms of (sn)(s_n). Let (sn)(s_n) converge to ss, with sn≠ss_n \ne s for all large nn. The transformation TT accelerates the convergence of (sn)(s_n) if (tn)(t_n) converges to ss and lim⁡n→∞tn−ssn−s=0.\lim_{n \to \infty} \frac{t_n - s}{s_n - s} = 0. Series acceleration means applying such a transformation to the sequence of partial sums of a convergent series.

Remarks

Example: for sn=2−ns_n = 2^{-n} with limit 00, the transformation tn=snsn+1=2−2n−1t_n = s_n s_{n+1} = 2^{-2n-1} gives tn/sn=2−n−1→0t_n / s_n = 2^{-n-1} \to 0, so it accelerates this sequence. Non-example: the shift tn=sn+1t_n = s_{n+1} gives tn/sn=12t_n / s_n = \frac12 for the same sequence. The error is halved, but the ratio does not tend to 00, so this is not acceleration in the sense above. A given transformation accelerates some sequences and not others.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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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 levelSeriesdefinitionPartial sum, definition, no evidence levelPartial sumdefinitionSeries acceleration, definition, no evidence levelSeries accelerationdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Sequence definition
  2. Real sequence definition
  3. Limit of a sequence definition
  4. Series definition
  5. Partial sum definition
  6. Series acceleration definition

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Series_acceleration
About this page
Tag90ANMX55DP
URIm2://90ANMX55DP
Kinddefinition
WritingW0 stub
MSC65B10, 65B05, 40A25
DefinesSeries acceleration
UsesLimit of a sequence, Partial sum, Series
Changed2 November 2026

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