First and second Born (e, 2e) calculations are presented for excitation-ionization of ground-state helium to He + (n = 2). Results for ionization to the ground-state ion He + (1s) are also given. The physical content of the approximations is discussed, in particular, the two-step mechanism which appears in the second-order term for excitation-ionization. The second Born term is calculated in the closure approximation using a new numerical method based on prolate spheroidal coordinates. Comparison is made with absolute experimental data from Paris and Rome in very asymmetric coplanar geometry-scattered electron energies of 5500, 1500 and 570 eV and ejected electron energies of 5, 10, 20, 40 and 75 eV. For excitation-ionization the second Born approximation generally gives improved agreement with the experimental data in the recoil region and second-order effects are found to be still significant at 5500 eV. The importance of the second-order term decreases with increasing ejected energy for the cases studied here. † Whereas the 2s → 1s transition is not 'optically allowed', the 2s state can decay radiatively to 1s by emission of two photons. These photons will have a distribution of energies and will therefore not appear as a single spectral line like the optically allowed 2p → 1s decay.‡ And also with collisions of other projectiles such as Li q+ , B q+ , C q+ , H + 2 and H +