  Sequence transformations accomplish an acceleration of convergence or a
summation in the case of divergence by detecting and utilizing regularities of
the elements of the sequence to be transformed. For sufficiently large indices,
certain asymptotic regularities normally do exist, but the leading elements of
a sequence may behave quite irregularly. The Gaussian hypergeometric series 2F1
(a, b; c; z) is well suited to illuminate problems of that kind. Sequence
transformations perform quite well for most parameters and arguments. If,
however, the third parameter $c$ of a nonterminating hypergeometric series 2F1
is a negative real number, the terms initially grow in magnitude like the terms
of a mildly divergent series. The use of the leading terms of such a series as
input data leads to unreliable and even completely nonsensical results. In
contrast, sequence transformations produce good results if the leading
irregular terms are excluded from the transformation process. Similar problems
occur also in perturbation expansions. For example, summation results for the
infinite coupling limit k_3 of the sextic anharmonic oscillator can be improved
considerably by excluding the leading terms from the transformation process.
Finally, numerous new recurrence formulas for the 2F1 (a, b; c; z) are derived.
