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Concept
Series acceleration
Summary
In mathematics, series acceleration is one of a collection of sequence transformations for improving the rate of convergence of a series. Techniques for series acceleration are often applied in numerical analysis, where they are used to improve the speed of numerical integration. Series acceleration techniques may also be used, for example, to obtain a variety of identities on special functions. Thus, the Euler transform applied to the hypergeometric series gives some of the classic, well-known hypergeometric series identities.
Given a sequence
having a limit
an accelerated series is a second sequence
which converges faster to than the original sequence, in the sense that
If the original sequence is divergent, the sequence transformation acts as an extrapolation method to the antilimit .
The mappings from the original to the transformed series may be linear (as defined in the article sequence transformations), or non-linear. In general, the non-linear sequence transformations tend to be more powerful.
Two classical techniques for series acceleration are Euler's transformation of series and Kummer's transformation of series. A variety of much more rapidly convergent and special-case tools have been developed in the 20th century, including Richardson extrapolation, introduced by Lewis Fry Richardson in the early 20th century but also known and used by Katahiro Takebe in 1722; the Aitken delta-squared process, introduced by Alexander Aitken in 1926 but also known and used by Takakazu Seki in the 18th century; the epsilon method given by Peter Wynn in 1956; the Levin u-transform; and the Wilf-Zeilberger-Ekhad method or WZ method.
For alternating series, several powerful techniques, offering convergence rates from all the way to for a summation of terms, are described by Cohen et al.
A basic example of a linear sequence transformation, offering improved convergence, is Euler's transform.
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