Padé Approximants

You may have a Taylor series which is slowly convergent or downright divergent, in which case you might want to try a Padé approximant. This is essentially a method which approximates a function in terms of the ratio of two polynomials, of orders N,M. A particularly useful representation is in continued fractions; the (N+1)th member of the Padé sequence:
pade_1
for J >= 0 is given by:
pade_2
Bender & Orszag in their book give an algorithm for constants c, I’ve worked out the first few by hand for you. There’s no such thing as a free lunch; you might find a converged value for your function, but it’s at the expense of nasty analysis in finding progressively higher orders for c
pade_3

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