The boundary-layer equations are solved numerically for mainstreams \[ U(x) = x(1-x^2)^{-\alpha}\quad {\rm and}\quad U(x) = (1-x)^{-\alpha}, \] which are both O((1 − x)−α) near x = 1. Series expansions are derived near x = 1. For α > 1, where, for the similarity solution at x = 1, the outer boundary condition is approached through exponentially small terms, a straightforward expansion in powers of 1 − x is possible. For 0 < α < 1, where the decay is only algebraic (Brown & Stewartson 1965), the outer boundary condition cannot be satisfied even with algebraic decay by the higher-order terms in the series and this must be regarded as only an inner expansion. An outer expansion is required which matches with this inner expansion and which approaches the outer boundary condition with exponential decay. For α = 1, the decay is exponential, but not of the same form as for α > 1, and again the outer boundary condition cannot be attained by the higher-order terms in the series. An outer expansion for this case is also derived.