  The Gaver-Stehfest algorithm is widely used for numerical inversion of
Laplace transform. In this paper we provide the first rigorous study of the
rate of convergence of the Gaver-Stehfest algorithm. We prove that
Gaver-Stehfest approximations converge exponentially fast if the target
function is analytic in a neighbourhood of a point and they converge at a rate
$o(n^{-k})$ if the target function is $(2k+3)$-times differentiable at a point.
