  We present a systematic procedure to extract the perturbative series for the
ground state energy density in the Lieb-Liniger and Gaudin-Yang models,
starting from the Bethe ansatz solution. This makes it possible to calculate
explicitly the coefficients of these series and to study their large order
behavior. We find that both series diverge factorially and are not Borel
summable. In the case of the Gaudin-Yang model, the first Borel singularity is
determined by the non-perturbative energy gap. This provides a new perspective
on the Cooper instability.
