Lewin, Jonathan W.

Lewin, Myrtle

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Kennesaw State UniversityDigitalCommons@Kennesaw State UniversityFaculty Publications12-1988A Simple Test for the nth Term of a Series toApproach ZeroJonathan W. LewinKennesaw State University, jlewin@kennesaw.eduMyrtle LewinAgnes Scott CollegeFollow this and additional works at: https://digitalcommons.kennesaw.edu/facpubsPart of the Mathematics CommonsThis Article is brought to you for free and open access by DigitalCommons@Kennesaw State University. It has been accepted for inclusion in FacultyPublications by an authorized administrator of DigitalCommons@Kennesaw State University. For more information, please contactdigitalcommons@kennesaw.edu.Recommended CitationLewin, J., & Lewin, M. (1988). A simple test for the nth term of a series to approach zero. The American Mathematical Monthly,95(10), p. 942. THE TEACHING OF MATHEMATICS EDITED BY MELVIN HENRIKSEN AND STAN WAGON A Simple Test for the nth term of a Series to Approach Zero JONATHAN LEWIN Department of Mathematics, Kennesaw College, Marietta, GA 30061 MYRTLE LEWIN Department of Mathematics, Agnes Scott College, Decatur, GA 30030 Using Stirling's formula, one may see at once that if an = (2n)!/4n(n !)2, then an is of the order of 1/ Vn, and one may conclude from the alternating series test that the series E( - l)nan is conditionally convergent. At an elementary level, however, the convergence of the latter series may be a little more difficult to obtain. Since an+11an = (2n + 1)/(2n + 2) < 1 for each n, it is clear that the sequence (an) is decreasing, but it is not immediately obvious within the environment of a typical calculus course that an--* 0 as n -- oo. For this purpose, one might use the following simple result which takes a leaf out of the theory of infinite products: THEOREM. Suppose (an) is a decreasing sequence of positive numbers and for each natural number n, define bn = 1 - an+1/an. Then the sequence (an) converges to zero if and only if the series Ebn diverges. Proof. We note first that unless bn -* 0 as n -- oo, both of the series Ybn and E log(l - bn) diverge. On the other hand, if bn 0, then bn/( - log(l - bn)) -* 1 as n -4 oo, and it follows from the limit comparison test that 2bn diverges if and only if ? log(l - bn) diverges. We note also that since 0 < bn < 1, we have log(l - bn) < 0 for each n. Now since 1 - bn = an+ /an for every n, it is clear that an = a1(1 - bl) (1 - b2)(1- b3) ... (1 - bn-1) for each n > 2, and we therefore conclude that an 0 iff log an - -oo iff log a, + Ei log(l - bi) - o- iff ,log(l - bn) diverges iff Ebn diverges. Returning now to the above example, we see that bn= 1/(2n + 2) for each n, and the obvious divergence of :2bn implies that an -* 0. The same technique gives an easy proof of the convergence of such series as -((- 1)'n/enn!), and the series E(t) of binomial coefficients with a > -1. Universal Topological Spaces K. D. MAGILL, JR. Department of Mathematics, SUNY, Buffalo, NY 14214 Let U = {a, b, c } and let Y1 = { U, 4, {a}}. It has been known for a long time that U with the topology 31 is a universal topological space in the sense that any topological space whatsoever is homeomorphic to a subspace of some topological 942This content downloaded from 130.218.13.44 on Thu, 08 Jun 2017 22:28:03 UTCAll use subject to http://about.jstor.org/terms

A Simple Test for the nth Term of a Series to Approach Zero

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A Simple Test for the nth Term of a Series to Approach Zero

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