We discuss a finite rectangular well of a depth λ2 as a perturbation for the infinite one with λ as a perturbation parameter. In particular, we consider a behavior of energy levels in the well as functions of complex λ. It is found that all the levels of the same parity are defined on infinitely sheeted Riemann surfaces whose topological structures are described in detail. These structures differ considerably from those found in models investigated earlier. It is shown that perturbation series for all the levels converge what is in a contrast with the known results of Bender and Wu. The last property is shown to hold also for the infinite rectangular well with the Dirac delta barrier as a perturbation considered earlier by Ushveridze.