The basic step in the construction of a sequence transformation is the
assumption that the elements 
of the sequence to be transformed can
for all indices 
be partitioned into a limit or
antilimit s and a remainder 
according to
If a sequence 
satisfying this assumption converges, it is at
least in principle possible to make the remainders negligible by
increasing the index n as much as necessary. However, many sequences
are known which converge so slowly that it is practically impossible to
obtain sufficiently accurate approximations to their limits by
increasing the index. Moreover, in the case of a divergent sequence,
increasing n does not help at all since it normally only aggravates
divergence.
As an alternative, one could try to improve the convergence of a
sequence 
or to sum it in the case of
divergence by computing sufficiently accurate approximations to the
remainders 
, and to eliminate these
approximations from the sequence elements.
The Euler-Maclaurin formula (see for instance section 13 of [8], section 8 of [36], or [37]) is an example for this approach: The remainder
of the partial sum (1.1) is replaced by an integral, for which
asymptotic approximations as 
are derived. The resulting
approximation to the remainder is then eliminated from the partial sum
, yielding a better approximation 
to the limit s of the
infinite series 
.
The major drawback of the Euler-Maclaurin formula is that it can only be
applied if the terms 
of the series are known explicitly as
functions of the index n, and if the resulting integral for the
truncation error has a sufficiently simple structure to allow further
manipulations. Consequently, it cannot be applied if only the
numerical values of the terms of a series are known.
Sequence transformations also try to compute approximations to the remainders and to eliminate them from the sequence elements. However, they differ from the Euler-Maclaurin formula in one very essential respect: They require only relatively little knowledge about the n-dependence of the remainders of the sequence to be transformed. Consequently, they can be applied in situations in which apart from the numerical values of the sequence elements only comparatively little is known.
Since the Euler-Maclaurin formula utilizes a lot of information, it
should in those cases, in which it can be applied, give better results
than sequence transformations, which utilize much less information. This
is a price which has to be paid for the fact that sequence
transformations are much more general numerical tools which can be
applied to a very wide range of problems
.
Normally, sequence transformations eliminate only approximations of the
remainders. Thus, the elements of the transformed sequence 
can also be partitioned into the limit
or antilimit s and a transformed remainder according to
The transformed remainders 
will in
general be different from zero for all finite values of n. However,
convergence is accelerated if the transformed remainders 
vanish more rapidly than the original
remainders 
as 
according to
and a divergent sequence is summed if the transformed remainders vanish
at all as 
.
As mentioned above, a sequence transformation tries to obtain a better
approximation to the limit or antilimit s by eliminating 
as
effectively as possible from 
. Of course, this is easier said than
done. The remainders 
of a sequence 
are in general unknown, and their determination is
normally not easier than the determination of the limit or antilimit s
of this sequence.
Accordingly, the exact remainders are normally either unknown or
not easily accessible, and a straightforward elimination is not
possible. Nevertheless, at least some structural and asymptotic
information about the dependence of the remainders 
on the index
n is often available.
First, I will try to clarify by means of some simple examples the term structural and asymptotic information. As the most simple example, let us consider the geometric series. Its truncation error is given by
Another example is the Euler integral
and its associated asymptotic series, the Euler series
which diverges quite strongly for every 
. It can be shown that
the truncation error of the Euler series satisfies (see for instance
Theorem 13-1 of [23])
For 
, the integral on the right-hand side is positive and
bounded in magnitude by the first term not included in the partial sum:
Thus, for 
the truncation error of both the geometric series
and the Euler series possesses the same sign as the first term not
included in the partial sum, and is bounded in magnitude by this term.
As is well known, the first term not included in the partial sum is a
simple estimate for the truncation error of a convergent series with
strictly alternating terms (see p. 259 of [1]). Moreover, it is
also the best simple estimate for the truncation error of a strictly
alternating nonterminating hypergeometric series 
with 
, which diverges quite strongly
for every 
(see Theorem 5.12-5 of [35]).
It will be shown later that powerful sequence transformations can be constructed on the basis of the assumption that the truncation error of the infinite series can be estimated by the first term not included in the partial sum.
The truncation error of the series for the Riemann zeta function, which is notorious for its bad convergence, satisfies (see p. 20 of [12])
where
Here, 
is a Bernoulli number. This representation of the
truncation error can be obtained from the Euler-Maclaurin formula
(compare example 3.2 on p. 292 of [36]).
In the case of the Riemann zeta function, the first term not included in
the partial sum would not be suited as an estimate for the truncation
error. However, the first term not included in the partial sum
multiplied by 
would be a simple estimate for the truncation
error (2.10) (see p. 382 of [38]).
Structural and asymptotic information on the n-dependence of the truncation errors, which need not be very detailed, can be used for the construction of sequence transformations. Since it would be futile to try to eliminate a remainder with a completely unknown and arbitrary n-dependence, a sequence transformation has to make either implicitly or explicitly some assumptions.
As a consequence, a sequence transformation will only work well if the actual behavior of the remainders is in sufficiently close agreement with the assumptions made.
Of course, this also implies that a sequence transformation, which makes specific and detailed assumptions about the remainders, may fail to accelerate convergence or to produce a convergent result in the case of divergence, if it is applied to a sequence with remainders of a sufficiently different behavior.
Thus, the efficiency of a sequence transformation for certain types of sequences and its inefficiency or even nonregularity in the case of other sequences are intimately related.
