Summary: The Borel method of summation is applied to Rayleigh-Schrödinger expansions with nonzero radius of convergence. The region of Borel summability in the complex plane of the interaction parameter is determined, and the effect of intruder states is discussed. For intruder states which are strongly coupled to the ground state the perturbation series can be Borel summable much further out than the radius of convergence. The application of Padé approximants to the Borel series is compared to the use of Padé approximants directly on the original series. Numerical applications are made to a simple model consisting of a \(2\times 2\) matrix Hamiltonian.