  We compare two alternative expansions for finite attractive wells. One of
them is known from long ago and is given in terms of powers of the strength
parameter. The other one is based on the solution of the equations of the
Rayleigh-Schr\"{o}dinger perturbation theory in a basis set of functions of
period $L$. The analysis of exactly solvable models shows that although the
exact solution of the problem with periodic boundary conditions yields the
correct result when $L\rightarrow \infty$ the coefficients of the series for
this same problem blow up and fail to produce the correct asymptotic expansion.
