All sequences (series) in this section are sequences (series) of real or complex numbers.

A transformation of a convergent sequence $\{s_n\}$ with limit $\sigma$ into a sequence $\{t_n\}$ is called limit-preserving if $\{t_n\}$ converges to the same limit $\sigma$.

The transformation is accelerating if it is limit-preserving and if

Similarly for convergent series if we regard the sum as the limit of the sequence of partial sums.

It should be borne in mind that a sequence (series) transformation can be effective for one type of sequence (series) but may not accelerate convergence for another type. It may even fail altogether by not being limit-preserving.

If $S={\sum }_{k=0}^{\mathrm{\infty }}{(-1)}^k{a}_k$ is a convergent series, then

provided that the right-hand side converges. Here $\mathrm{\Delta }$ is the forward difference operator:

Thus

Euler’s transformation is usually applied to alternating series. Examples are provided by the following analytic transformations of slowly-convergent series into rapidly convergent ones:

This transformation is accelerating if $\{s_n\}$ is a linearly convergent sequence, i.e., a sequence for which

When applied repeatedly, Aitken’s process is known as the iterated ${\mathrm{\Delta }}^2$-process. See Brezinski and Redivo Zaglia (1991, pp. 39–42).

Shanks’ transformation is a generalization of Aitken’s ${\mathrm{\Delta }}^2$-process. Let $k$ be a fixed positive integer. Then the transformation of the sequence $\{s_n\}$ into a sequence $\{t_{n,2k}\}$ is given by

where $H_m$ is the Hankel determinant

The ratio of the Hankel determinants in (3.9.9) can be computed recursively by Wynn’s epsilon algorithm:

Then $t_{n,2k}={\epsilon }_{2k}^{(n)}$. Aitken’s ${\mathrm{\Delta }}^2$-process is the case $k=1$.

If $s_n$ is the $n$th partial sum of a power series $f$, then $t_{n,2k}={\epsilon }_{2k}^{(n)}$ is the Padé approximant ${[(n+k)/k]}_f$ (§3.11(iv)).

For further information on the epsilon algorithm see Brezinski and Redivo Zaglia (1991, pp. 78–95).

We give a special form of Levin’s transformation in which the sequence $s=\{s_n\}$ of partial sums $s_n={\sum }_{j=0}^n{a}_j$ is transformed into:

where $k$ is a fixed nonnegative integer, and

Sequences that are accelerated by Levin’s transformation include logarithmically convergent sequences, i.e., sequences $s_n$ converging to $\sigma$ such that

For further information see Brezinski and Redivo Zaglia (1991, pp. 39–42).

In Weniger’s transformations the numbers $c_{j,k,n}$ in (3.9.13) are chosen as follows:

or

where ${\left(a\right)}_0=1$ and ${\left(a\right)}_j=a(a+1)\mathrm{\cdots }(a+j-1)$ are Pochhammer symbols (§5.2(iii)), and the constants $\beta$ and $\gamma$ are chosen arbitrarily subject to certain conditions. See Weniger (1989).

For examples and other transformations for convergent sequences and series, see Wimp (1981, pp. 156–199), Brezinski and Redivo Zaglia (1991, pp. 55–72), and Sidi (2003, Chapters 6, 12–13, 15–16, 19–24, and pp. 483–492).

For applications to asymptotic expansions, see §2.11(vi), Olver (1997b, pp. 540–543), and Weniger (1989, 2003).