Say you are only able to obtain a limited number of terms in a power series, as each term is increasingly difficult to obtain (ex. virial equation of state). To explore possible analytic continuations, you need an infinite series representation. A rational function is one way to prescribe this, although it is probably not the most general. Anyways, this lightweight code has helped me to quickly try different Pades to investigate how they extend the region of analyticity of the truncated series.
A similar code has be written to construct differential approximants to truncated series, following the method of Guttmann & Joyce (1972). I'll post that if there is interest.