step, pest, pset, tesp, ptes, september.

Their basic idea is to extract hidden information contained in partial sums of a specific slowly
convergent or divergent series, and to use that information in order to make a qualified estimate
about new (usually higher order) partial sums which eventually converge to some limit. In many
cases, this “qualified estimate” leads to spectacular numerical results which represent a drastic
improvement over a term-by-term summation of the original series, even if the series is formally
convergent. For further discussion it is useful to consider a sequence {{sn}} = {{s0, s1, . . .}}
with elements sn or the terms an = sn − sn−1 of an infinite series. Sequence transformations are
important tools for the convergence acceleration of slowly convergent sequences or series and
also for the summation of divergent series. The basic idea is to construct from a given sequence
{{sn}} a new sequence {{s′n}} = T({{sn}}) where each s′n depends on a finite number of ele
ments sn1, . . . , snm. Often, the sn are the partial sums of an infinite series. The aim is to find a
transformation T such that {{s′n}} converges faster than sn or, after all, it is capable to sum
{{sn}}. A common approach is to rewrite sn as sn = s + Rn (1) where s is the limit (or anti-limit
in the case of divergence) and Rn is the remainder. The aim then is to find a new sequence {{s′n}}
such that s′n = s + R′n , R′n /Rn → 0 for n → ∞ (2) Thus, the sequence {{s′n}} converges faster
to the limit s (or diverges less violently) than {{sn}}.