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

On two perturbation series for short-range attractive potentials