In the case of a Stieltjes series, an integral representation for the truncation error can be obtained quite easily. We only have to insert the relationship
into the integral representation (6.2) and do the moment integrals according to eq. (6.3). This yields
If the argument z is positive, the truncation error of a Stieltjes series has the same sign as the first term not included in the partial sum and is bounded in magnitude by this term:
This estimate can easily be generalized to arbitrary complex arguments
z that are not on the cut 
[23, Theorem 13-2,].
Two cases have to be distinguished depending on 
. If
,
and if 
,
These estimates show that the first term not included in the partial sum
is a simple estimate for the truncation error of a Stieltjes series, as
long as z is not on the cut 
. However, the first term
not included in the partial sum is nothing but the remainder estimate
(5.3) which is the basis of 
, eq. (5.6), and 
, eq. (5.7).
The results in Tables 2 and 3 show that the ``trivial'' remainder estimate
is able to produce very good summation results. Nevertheless, it would
be interesting to investigate whether and how more refined remainder
estimates 
for Stieltjes series would influence the
performance of the ``parent'' sequence transformations 
, eq. (4.5), and 
, eq. (4.11).
An obvious generalization of the ``trivial'' remainder estimate
(7.6) would be the following asymptotic expansion of the
truncation error integral in eq. (7.2) in powers of
:
I do not know whether or under which conditions such an expansion can be constructed. In any way, it would certainly be interesting if at least some of the coefficients A, B, C, ... could be obtained in closed form.
Alternatively, one could also assume that the truncation error integral in eq. (7.2) can be expressed as a factorial series of the following kind:
Again, I do not know whether or under which conditions such an expansion
can be constructed. However, I would be happy if at least some of the
coefficients 
, 
, 
, ... could be obtained.
In asymptotics, it is quite common to analyze predominantly those
remainder terms of asymptotic series which are minimal in magnitude
(compare for instance the remainder term of the asymptotic series for
the exponential integral on p. 523 of Olver's book [36]). This
means that the argument z and the index n of the remainder term are
not independent. This approach is not possible in the case of the two
asymptotic expansions (7.7) and (7.8). The
reason is that the sequence transformations 
, eq. (4.5), and 
, eq. (4.11), require as input data the sequence
elements 
, 
, ... , 
and the remainder estimates
, 
, ... , 
. Since whole strings
of remainder estimates are needed, the argument z and the index n
have to be independent. Consequently, the argument z in eqs.
(7.7) and (7.8) is essentially arbitrary
apart from the restriction that z must not be on the cut 
. Moreover, if the coefficients A, B, C, ... and 
, 
,
, ... in eqs. (7.7) and (7.8) exist,
they will depend explicitly on z.
A different, but nevertheless very important class of problems arises if
the argument z is a negative real number, i.e., if the argument is on
the cut 
. In this case, the relationship
has to be used in the integral representation (6.2). If the moment integrals are done according to eq. (6.3), we obtain:
Here, the integrals are to be interpreted as principal value
integrals. Consequently, 
with z > 0 will have a
nonzero imaginary part, although z and the corresponding measure
are real.
If we would want to sum the divergent sequence of partial sums
by the sequence transformations 
, eq. (4.5), and 
, eq. (4.11), we would need an estimate for the
truncation error integral in eq. (7.10), which permits the
summation of a divergent series of positive terms to something with a
nonzero imaginary part.
Special cases of the problems formulated in this section were already treated in the literature. For example, in section 21 of Dingle's book [69], the remainder terms
and
of the asymptotic series for the exponential integral were investigated.
