A sequence of limit tests for the convergence of series

Abstract

In this paper, we shall develop a sequence of limit tests for the convergence and divergence of infinite series of positive terms which is similar in form to the De Morgan and Bertrand sequence but involves the ratio of two successive values of the test ratio rather than the test ratio itself. The proof will be based on the following integral test by R. W. Brink: and if R(x) is a function such that R(n) =R nj and such that R(x) *zR(x') when x' >x, a necessary and sufficient condition for the convergence of the series ^n°LoWn is the convergence of the integral I exp S--I I log R(x) dxdx> cSince a finite number of terms does not affect convergence or divergence, the conditions of Theorem VI need hold only for n greater than some fixed number v, in which case zero is to be replaced by v as a lower limit of integration.The foregoing theorem admits a generalization similar to that given by C. T. Rajagopal 3 in the case of another theorem of Brink's. 4 However, Brink's Theorem VI is sufficient for the purposes of the present paper.LEMMA. Let \u n \ and \uù } be sequences of positive terms with ratios r n =u n+ i/u n , R n = r n +i/ r n, rû =Un + i/un , and Rn =^+iA«» such that \\m n ^oo r n = lim nH>00 r n ' = 1.

A new test for convergence of positive series

“…We show how the new definitions lead to new convergence/divergence tests. For the undecided cases, we generalize an old result of Martin [11]. In the final remarks, we provide some one-sided results.…”

“…c) We prove a generalization of an old result of Martin [11]. Let ln (0) z = z, ln (1) z = ln z and ln (k+1) z = ln ln (k) z for k = 1, 2, .…”

A new test for convergence of positive series

“…Similarly we have c) We prove a generalization of an old result of Martin [11]. Let ln (0) z = z, ln (1) z = ln z and ln (k+1) z = ln ln (k) z for k = 1, 2, ...…”

Necessary and sufficient conditions for the convergence of positive series

“…In the present note, we establish necessary and sufficient conditions for convergence of positive series that generalize the original version of the extended Bertrand-De Morgan test [1,9]. The first theorem on a necessary and sufficient condition for convergence of series was obtained by Cauchy [7], widely known as Cauchy's convergence test.…”

“…The extended Bertrand-De Morgan test is the last test in this hierarchy. It was originally established in [9]. An elementary proof of this test, its connection with Kummer's test, as well as its application to birth-and-death processes is given in [1].…”