The Neumann series is a well-known technique to aid the solution of uncertainty propagation problems. However, convergence of the Neumann series can be very slow, often making its use highly inefficient. In this article, a fast convergence parameter () convergence parameter is introduced, which yields accurate and efficient Monte Carlo–Neumann (MC-N) solutions of linear stochastic systems using first-order Neumann expansions. The convergence parameter is found as a solution to the distance minimization problem, for an approximation of the inverse of the system matrix using the Neumann series. The method presented herein is called Monte Carlo–Neumann with convergence, or simply the MC-N method. The accuracy and efficiency of the MC-N method are demonstrated in application to stochastic beam-bending problems.