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.