A property of power series with positive coefficients

The following theorem is suggested by a problem in the theory of probability .' Let pk be a sequence of non-negative numbers for which Eó pk =1, and let m = Ei kpk <_ "0. Suppose further that P(x) _ E p kx k 0 is not a power series in xl for any integer t > 1. Then 1-P(x) has no zeros in the circle I x I < 1, and the series U(x) _ u • k x k 1-P(x) 0 has the property lim u n = 1/m. n-(If m= oo, we define 1/m to be zero .) We shall first give a proof in case m <-. The method used is not elementary, but yields somewhat more information than stated in the theorem. Later in this paper an elementary proof is given, valid for both m<-and m =-. We suppose that m < o o. Let rn = pk, R(x) _ r n xn. k=n+1 0 Then m = E0 rk and (2) 1-P(x) _ (1-x)R(x). Since m < oo the power series for R(x) converges absolutely and uniformly in I x I <=1. We claim that R(x) has no zeros for I x I <=1. For I xI < 1 this is clear from (2), since P(x) has positive coefficients Received by the editors March 1, 1948. 1 To be published elsewhere. The theorem and method of the present paper were extended to the continuous case by D. Blackwell, A renewal theorem, Duke Math .