Welch [22] stated the Behrens-Fisher problem as a partial differential equation of infinite order and described how to obtain an exact solution of it by a series approach in reciprocal numbers of degrees of freedom. This solution gives the limit of the critical region of the Behrens-Fisher test variable as a function that only depends on the empirical variance ratio. However, Linnik [11] showed that such a function cannot be continuous, and up to now, it has not yet been commented upon that this contradicts Welch’s approach, whose solution is postulated to be infinitely often differentiable. This paper tries to dissolve this contradiction on the basis of the Welch-Aspin test, which uses the expansion of Welch’s series approach up to the fourth order. It becomes plausible that the convergence radius of Welch’s series is zero, so Welch’s approach does not provide an exact solution, and this is conform with Linnik’s non-existence theorem. The investigation of the error probability of the first kind shows the accuracy of the Welch-Aspin test, but also indicates that developing too high orders could deteriorate the results.