Airey's Converging Factor.
Asmptotic series for the calculation of functions, for values of the argument numerically >1, start off with terms whose numerical values decrease but, at a certain stage, the terms begin to increase in numerical value and must be ignored. At this stage, there may be two adjacent terms of equal numerical value; when the least term of the asymptotic series is spoken of, it is in reference to the first of these two terms. The sum of the initial terms of the asymptotic series up to, and including, the least term often furnishes a fair approximation to the desired value of the function being evaluated. It was early observed by computers that if the terms of the asymptotic series alternate in sign, this approximation was often improved by replacing the least term by its half. The factor by which the least term of the asymptotic series must be multiplied so that the true value of the function being evaluated is obtained by addition of this modified least term to the remaining initial terms of the asymptotic series is known as the converging factor for the asymptotic series. The converging factor for the asymptotic series involved in the calculation of the exponential integral, for large negative values of the argument, was given as a power series in the reciprocal of the argument by Airey; the first term of this series is (1/2). A method for the determination of the coefficients of this series is given.
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