The sequence transformations (4.5) and (4.11) differ from
other sequence transformations like Wynn's 
[13] or
Brezinski's 
[15] algorithm in one very important
aspect: They require as input data not only the elements of a sequence
, but also explicit remainder estimates 
. The explicit incorporation of the
information contained in the remainder estimates makes these
transformations potentially very powerful. However, this is also the
major potential weakness of these transformations. If remainder
estimates can be found such that the products 
provide
good approximations to the remainders of the sequence to be transformed,
sequence transformations like (4.5) or (4.11) should work
very well. If, however, good remainder estimates cannot be found,
sequence transformations of that kind will perform poorly. Consequently,
a fortunate choice of the remainder estimates in eqs. (4.5) and
(4.11) is of utmost importance for the success or failure of a
convergence acceleration or summation process.
The difference operator 
is linear. Consequently, the effect of
the general sequence transformation (4.4) on an arbitrary
sequence 
with limit or antilimit s can be
expressed as follows:
converges to s if the ratio on the right-hand side can be made small.
This will be the case if the weighted difference operator 
annihilates 
more effectively than
. Thus, remainder estimates have to be found such that
depends on n sufficiently strongly, whereas 
depends on n only quite weakly:
This asymptotic condition can be the starting point for the construction
of remainder estimates. However, such an asymptotic condition does not
determine the remainder estimates 
for a given sequence
uniquely. Instead, it is at least in principle possible to
find for a given sequence an infinite variety of different remainder
estimates which all satisfy the asymptotic condition
(5.2), and which may perform quite differently in
convergence acceleration and summation processes [24].
On the basis of heuristic and asymptotic arguments, Levin [38] suggested for sequences of partial sums (1.1) several simple explicit remainder estimates. The use of these remainder estimates in (4.5) yields Levin's u, t, and v transformation, respectively [38]. Later, Smith and Ford [46] suggested the remainder estimate
which is -- as remarked before -- actually the most obvious simple estimate for the truncation error of a convergent or divergent alternating series.
A more detailed discussion of the properties of these remainder estimates, some generalizations, additional heuristic motivation, and a description of the types of sequences, for which these estimates should be effective, can be found in section 7 of [23]. The main advantage of Levin's remainder estimates [38] and of the remainder estimate (5.3) is that they can be used in situations in which only the numerical values of a few terms of a slowly convergent or divergent series are known.
Levin's simple remainder estimates [38] and the remainder estimate (5.3) are not restricted to Levin's transformation, but can also be used in the sequence transformation (4.11) (see section 8.4 of [23]). In spite of their simplicity, the use of these remainder estimates in eqs. (4.5) and (4.11) leads to powerful sequence transformations [23,28,29,30,31,32,33,34,46,47], which are obviously nonlinear and also nonregular (see sections 7.3, 8.4, and 12.3 of [23]). The theoretical convergence properties of the sequence transformations (4.5) and (4.11) and their variants were analyzed in articles by Sidi [64,65,66], in sections 12 - 14 of [23], in section 4 of [32], and also in section 5.7 of [33].
This report is primarily interested in the summation of certain strongly divergent power series. Consequently, only the remainder estimate (5.3) will be considered here explicitly. Its use in eqs. (4.5) and (4.11) yields the following sequence transformations [23, eqs. (7.3-9) and (8.4-4),]:
If these transformations are applied to partial sums (1.3)
of the power series for 
, rational functions 
and 
result with numerator and denominator polynomials of degrees
k+n and k, respectively [32, eqs. (4.26) and (4.27),]:
If the coefficients 
of the power series for 
are all
different from zero, the rational functions 
and 
satisfy for
the following error estimates [32, eqs. (4.28) and
(4.29),]
Consequently, the Taylor expansions of 
and 
at z = 0
reproduce the power series for 
up to terms of order k + n + 1:
These two expressions are formally very similar to the analogous
expression (1.8) for Padé approximants. As in the case
of Padé approximants, any difference in the numerical properties of
the rational approximants 
and
and of the partial sums
(1.3), which were used for their construction, must be
attributed to the truncation errors 
and
, respectively, which are in general
nonzero.
However, the truncation errors 
and
differ from the truncation error 
in eq. (1.8) in one very important
aspect: In particular for strongly divergent series, they seem to yield
much better approximations to the truncation errors of the formal power
series which are to be summed.
To demonstrate this, let us consider the well known asymptotic series for the exponential integral,
which diverges quite strongly for all 
.
In Table 2,Wynn's 
algorithm [13], and the sequence
transformations 
and
are applied to the partial
sums
of the asymptotic series (5.12) with 
.
Wynn's 
algorithm [13] computes Padé approximants
according to
if the partial sums (1.3) of the power series for 
are used as input data. In Table 2, the notation 
for the
integral part of x is used. Thus, the elements 
with 
, which occur in
the third column of Table 2, correspond to the following sequence of
Padé approximants:
The results in Table 2 show quite clearly that 
and 
, which use
explicit remainder estimates, are in the case of the asymptotic series
(5.12) for 
much more effective than
Padé approximants.
Another example, which demonstrates the principal advantages of sequence transformations using explicit remainder estimates, is the following series expansion for the logarithm:
For 
, the power series converges absolutely, for z =
1 it converges conditionally, and for 
it diverges.
However, as long as the argument z does not lie on the cut 
, the divergent series can at least in principle be summed.
In Table 3, the same transformations as in Table 2 are applied to the partial sums
of the power series for 
with z = 5. The partial sums in
column 2 of Table 3 show that the power series diverges quite strongly
for an argument as large as z = 5.
The results in Table 3 are qualitatively same as the results in Table 2:
Padé approximants are apparently able to sum the divergent series,
but the sequence transformations 
and 
, which use explicit
remainder estimates, are much more effective and clearly outperform
Padé approximants.
There exist connections between the theory of converging factors
(see for instance Airey [67], Miller [68], Sections 21 -
26 of Dingle [69], Neuhaus and Schottlaender [70],
and section 14 of Olver [36]) and the sequence transformations
and 
. In the theory
of converging factors, the truncation error 
of a divergent series
is expressed as the first term 
,
which was not included in the partial sum 
, multiplied by a
converging factor 
which is to be chosen in such a way that the
relationship
is satisfied. Hence, the sequence transformations 
and 
, which are based upon the remainder
estimate (5.3), essentially try to accomplish the determination
and elimination of the converging factor 
by purely numerical
means. The sequence transformation 
assumes that
can be expressed as a power series in 
according to
eq. (4.7) and tries to determine the k leading coefficients
, 
, 
, 
of the power series. Similarly,
assumes that 
can be expressed as
a factorial series according to eq. (4.13) and tries to
determine the k leading coefficients 
, 
, 
, 
of the factorial series.
