As already remarked, the asymptotic condition (5.2) can be
the starting point for the construction of remainder estimates 
. In my opinion, it should be possible to derive many interesting
results which could lead via the sequence transformations (4.5)
and (4.11) to powerful approximation schemes for functions
defined by slowly convergent or divergent series.
For instance, in [29] it was shown that the asymptotic series for the modified Bessel function of the second kind,
which diverges quite strongly for all arguments 
, can be summed efficiently by the sequence transformations
and 
. These two
transformations are based upon the remainder estimate (5.3),
which corresponds to the first term of the series not contained in the
partial sum. It would be interesting to find out whether and how well
the efficiency of the ``parent'' sequence transformations (4.5)
and (4.11) could be increased by better truncation error
estimates.
Special results of that kind would certainly be interesting. However, it would be much more interesting to construct remainder estimates which are generally valid for whole classes of functions. Of course, this would probably be much more difficult. Nevertheless, I hope that the derivation of general results could at least in the case of Stieltjes series be accomplished.
A function 
is called Stieltjes function if it can be
expressed as a Stieltjes integral according to
Here, 
is a positive measure on 
, which
assumes infinitely many different values on 
and which
has for all 
finite and positive moments 
defined by
The formal series expansion for 
, which need not be convergent,
is called a Stieltjes series if its coefficients 
are
moments of a positive measure 
on 
according
to eq. (6.3):
The classic example of a Stieltjes function with a strongly divergent Stieltjes series is the Euler integral (2.6) and its associated asymptotic series, the Euler series (2.7).
The series (5.16) for 
, which was summed in Table 3,
is also a Stieltjes series. This follows from the integral
representation
We only have to set 
for 
and 
for 
. The moments 
of this positive measure
are given by
Stieltjes functions and their associated Stieltjes series are very important in the theory of divergent series, since they possess a highly developed representation and convergence theory [25,71,26,72,73,74].
Moreover, Stieltjes functions and Stieltjes series are also of considerable importance in quantum mechanical perturbation theory. An example is the quartic anharmonic oscillator which is described by the Hamiltonian
Simon [75] could show that the perturbation series for an energy eigenvalue of the quartic anharmonic oscillator oscillator,
which diverges quite strongly for every nonzero coupling constant
, is the negative of a Stieltjes series.
If the positive measure 
corresponding to the Stieltjes
function 
could be determined directly from the Stieltjes moments
, the value of 
could at least in principle be
computed via the integral representation (6.2) even if the
Stieltjes series diverges. For instance, the measure corresponding to
the asymptotic series (6.1) for the modified Bessel function
can be found quite easily [29, eq. (2.25),].
Moreover, there is an extensive literature on moment problems and their
solution [76]. Necessary and sufficient
conditions, which for instance guarantee that Padé approximants are
able to sum a divergent Stieltjes series, have been known for long. An
example is the condition that all Hankel determinants formed from the
moments 
have to be strictly positive [26, Theorem
5.1.2,]. Unfortunately, the practical application of this and
similar other conditions is by no means simple, in particular if only
the numerical values of a finite number of Stieltjes moments 
,
, ..., 
are available.
However, there is a comparatively simple sufficient condition,
the so-called Carleman condition. If the moments 
satisfy
then the corresponding moment problem possesses a unique solution, and
the sequences 
of Padé approximants converge for j > -
1 to the value of the corresponding Stieltjes function as 
[77, Theorem 12.11f and Corollary 12.11h,]. It can be
shown that the Carleman condition is satisfied if the moments 
do
not grow faster than 
as 
, with C being
a suitable positive constant [78, Theorem 1.3,].
The Carleman condition (6.10) implies that the Euler series
(2.7) can be summed by the Padé approximants 
with fixed j > - 1. However, the numerical results in Tables 2 and 3
demonstrate convincingly that Padé summation need not be the most
efficient way of extracting useful numerical information from the terms
of a divergent Stieltjes series. Consequently, it should be worth while
to investigate also other techniques for the summation of divergent
Stieltjes series.
