Upper and lower limits for the expected number $n$ of observations required by a sequential probability ratio test have been derived in a previous publication [1]. The limits given there, however, are far apart and of little practical value when the expected value of a single term $z$ in the cumulative sum computed at each stage of the sequential test is near zero. In this paper upper and lower limits for the expected value of $n$ are derived which will, in general, be close to each other when the expected value of $z$ is in the neighborhood of zero. These limits are expressed in terms of limits for the expected values of certain functions of the cumulative sum $Z_n$ at the termination of the sequential test. In section 7 a general method is given for determining limits for the expected value of any function of $Z_n$.