A New Integral Test for the Convergence and Divergence of Infinite Series
Raymond W. Brink · Transactions of the American Mathematical Society · 1918
A new sequence of integral tests for the convergence and divergence of infinite series has been developed by the author.Some of the tests of this sequence, and the principle by which they may be discovered will be set forth by him in another article.In the present paper it is his desire to give a central one of these tests, together with some of its applications.This particular integral test appears to play the same rôle when the ratio of successive terms is explicitly known, that the Maclaurin-Cauchy test plays when the individual term is explicitly known.In testing a series of the form U0 + «1 + «2 + du Bois-ReymondJ called those tests that make use of the test ratio rn = Un+l!un, tests of the second kind to distinguish them from tests using the general term of the series un itself, which he called tests of the first kind.Similarly, the integral tests developed in this paper, which involve a function r (x) such that r(n) = r", may be called integral tests of the second kind; while the Maclaurin-Cauchy integral test, involving a function ti(x) for which u(n) = un, is an integral test of the first kind.Integral tests of the second kind thus apply to series for which a function is known that for successive integral values of the variable takes on the successive values of the ratio of one term to the preceding term.Such a series can be written in the following normal form: c -f ea0 + cd0 di + ca0 tu «2 + • • • , * Presented to the Society, April, 1916.