# 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:

for *J* >= 0 is given by:

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*…

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