In this paper, the Volterra series decomposition of a class of quadratic, time invariant single-input finite dimensional systems is analyzed. The kernels are given by a recursive sequence of linear PDEs in the time domain, and an equivalent algebraic recursion in the Laplace domain. This is used to prove the convergence of the Volterra series to a (possibly weak) trajectory of the system, to provide a practicable value for the radius of convergence of the input in L-infinity(R+) and to compute a guaranteed error bound in L-infinity(R+) for the truncated series. The result is then extended to MIMO systems. A numerical simulation is performed on an academic SISO example, to illustrate how easily the truncated Volterra series can be implemented.