Use this tag for questions about sequence transformations for improving the rate of convergence of a series.
Convergence acceleration is one of a collection of sequence transformations for improving the rate of convergence of a series or sequence.
Techniques for convergence acceleration are often applied in numerical analysis, where they are used to improve the speed of numerical integration. Convergence acceleration can also be used to obtain a variety of identities for special functions; for example, the Euler transform applied to the hypergeometric series yields hypergeometric-series identities.
Convergence acceleration can have interesting theoretical properties as well. For instance, in Apery's proof showing the irrationality of $\zeta(3)$, an important step was rewriting $$ \zeta(3) = \frac{5}{2}\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k^3 \binom{2k}{k}} $$The Ratio Test gives $1/4$ when applied to the new series, compared to $1$ using the usual definition $\zeta(s) = \sum_{n=1}^{\infty} n^{-s},$ $\Re(s)>1$.
