An Upper Bound on the Entropy Series

where 0 log 0 is taken as 0. Assume that the p~ are nonincreasing in n (reordering of the p~ does not affect H) . In a recent paper, Keilson (1971) showed that a sufficient condition for the series in (1) to be summable is that oo (i) Z1 P~ log n < ~ and (ii) P,.k ~ 1 / kp , as n ~ ~ for all fixed h = 1, 2 , . . . . I t will follow from the results given here that condition (i) alone is sufficient as well as necessary. Our main result is an upper bound on the entropy series (1) in terms of expectation of certain functions of the random variable X, where X takes the value n with probabi l i ty p~ (n = 1, 2,...).