A New General Method of Summing Divergent Series

Abstract

Foreword. The practice,in more advanced mathematics, of extending the meaning of terms already in use in elementary mathematics has so modified the meaning of the word sum that its friends in algebra could not be expected to recognize it without a reintroduction. Out of the multitude of usages of this word we are concerned here only with its use as the sum of an infinite series, meaning by infinite series an indicated sum of an unlimited number of terms, such as .9 .09 .009 .0009 In the study of infinite series (or just series as it is customary to call them), we mean by sum not the result of adding the terms as in algebra (an impossible feat since there are an unlimited number of them) but the limit of the algebraic sum of the first n terms as n increases beyond all bounds if this limit exists, in other words the limit of the sequence of its partial sums. (And this limit can often be found from the nature of the sequence). In the above example, the partial sums are .9 .99 .999 * * * and the limit of these partial sums is, of course, 1. As a further example consider the series 2+ + 8+ * *