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
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
, where they are used to improve the speed of
numerical integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations ...
. Series acceleration techniques may also be used, for example, to obtain a variety of identities on
special functions
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defined ...
. Thus, the
Euler transform applied to the
hypergeometric series gives some of the classic, well-known hypergeometric series identities.
Definition
Given a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
:
having a
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
:
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