  We explore in detail how analytic continuation of divergent perturbation
series by generalized hypergeometric functions is achieved in practice. Using
the example of strong-coupling perturbation series provided by the
two-dimensional Bose-Hubbard model, we compare hypergeometric continuation to
Shanks and Pad\'e techniques, and demonstrate that the former yields a
powerful, efficient and reliable alternative for computing the phase diagram of
the Mott insulator-to-superfluid transition. In contrast to Shanks
transformations and Pad\'e approximations, hypergeometric continuation also
allows us to determine the exponents which characterize the divergence of
correlation functions at the transition points. Therefore, hypergeometric
continuation constitutes a promising tool for the study of quantum phase
transitions.
