Abstract
To make the object of this work intelligible, it is necessary to recall a few facts concerning infinite series in general. Suppose we have a sequence u1, u2, u3, … un … (U) where u1, u2, &c., are analytical expressions constructed by a definite rule. Let sn=u1 + u2 + … + un; then we have a derived analytical sequence s1, s2, s3, … sn, …; (S) this is a definite analytical entity, and its properties are implicitly fixed by those of the former sequence. The expressions un, sn are, of course, functions of n; we may suppose, for simplicity, that, besides this, they involve, in addition to definite numerical constants, a single analytical variable, x. If we assign to x a numerical value, S becomes an arithmetical sequence, and three principal cases arise, according to the behaviour of sn when n increases indefinitely. If sn converges to a definite limit s we say that this is the sum of the series u1 + u2 + u3 + …, and write s = un; but the ultimate value of s may be either indeterminate or infinite. In the second case Σun has no definite meaning; in the third we may say, if we like, that Σun is infinite, but this infinite sum is not a quantity with which we can operate, and presents no special interest.
Leçons sur les Séries Divergentes.
Par Émile Borel. Pp. viii + 184. (Paris: Gauthier-Villars, 1901.) Price fr. 4.50.
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M., G. Leçons sur les Séries Divergentes . Nature 65, 2–3 (1901). https://doi.org/10.1038/065002a0
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DOI: https://doi.org/10.1038/065002a0