Series acceleration

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 ${\displaystyle \ell }$ 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 ${\displaystyle \ell }$.

...
Wikipedia