An Early Method for Summation of Series

Extract

1. The method given by Euler * for the transformation of series was used by him to obtain “sums” for various divergent series, and recently Knopp † has written extensively on the “Euler summation process” derived from this transformation. But it does not seem to have been remarked that an interesting variation of this method is given by Hutton in the first volume of his Tracts, published in 1812. In Tract 8, “The Valuation of Infinite Series,” written in 1780, Hutton gives an account of his method, claiming to have arrived at it independently of Euler’s work, and employing it to approximate to the sums of series whose terms alternate in sign. When the proposed series converges, we have an easy and rapid way of determining its sum, and in the case of a divergent series, a number is determined which would nowadays be recognized as a conventional sum. Very little attention seems to have been paid to Hutton’s paper and his name is apparently not mentioned by modern writers on this subject.‡ De Morgan § refers to the “remarkable method” of Hutton, but he applies it only to convergent series, and makes no reference to its application to oscillating series. This is noteworthy in view of the remarks made by De Morgan a t the end of his Chapter 19 on “The Transformation of Divergent Developments,” but it might be conjectured that even so acute a logician as De Morgan had not clearly perceived, at any rate at the time of writing his “Calculus,” that the word “sum” applied to any infinite series is being used in a conventional sense.