The use of moving sums (MOSUMS) of recursively calculated residuals (Hack1 [9]) is demonstrated for testing the constancy of the expected value of a series of T independent observations normally distributed with equal variances. The cases of known or unknown expected value are treated separately; in the situation of unknown variance the recursive estimation of the variance is suggested in the course of the sequential application of the test procedure. Conservative probability limits are constructed and an assessment of the effective error probabilities and the power of the method is given in simulation studies. Finally the use of the procedure is demonstrated in an example.