The summation of power series having positive coefficients

The paper develops a method for the summation of a series of positive terms given by Gismalla et al. [2]. Here we are concerned with the summation of power series ∑∞n = 1 μnxn where μn > 0 and 0 < x < 1, and x may be very close to 1 (but not equal to 1). Using ideas of Longman [4] we show how a bound can be obtained for the error when the process is curtailed at a particular stage. The method supposes that the coefficients in the summation are moments but, in practice, seems to work even when the coefficients are not identifiable as moments.