Our local choice was Matthews & Walker Mathematical Methods of Physics.
Always used to be popular with the turbulent flow brigade - often gave divergent power series that could only be tamed with Shank's. The pure mathematicians used to cringe at the abuse of method but they could not deny that the results it predicted matched experimental data well!
Works best on alternating series with poor convergence or divergence!
I was once very interested in extended convergence tricks and have used the rational approximation for Log(1+x) in anger several times.
Log(1+x) = x(6+x)/(6+4x) where -1/2 < x < 1
The other common one sometimes useful is
Sqrt(x) = (1+3x)/(3+x) where -1/2 < x < 2
Only really any good if you have fast hardware divide.
These tricks sometimes allow a rough polynomial approximation soluble analytically to provide a much better initial input guess for more rapid convergence of an iterative method like N-R.