ABSTRACT
The stern conventional wisdom, still taught to some physics students in their mathematics courses, is that if a power series fails to pass every test for convergence then people must have no further truck with it. If, however, it does pass, then it is 'one of the family' and all family members must be treated with equal respect. Various ways of rearranging the terms in a series can be tried to get quicker convergence for a convergent series. If the sequence of sums of the series is treated then the Aitken procedure is often effective. The really interesting effects, however, occur if a divergent series is treated by such methods, since there sometimes results a 'sum' for the series which on deeper examination turns out to be mathematically meaningful. It is the theory of Padé approximants and series of Stieltjes.
