A generalization of Bertrand's test

Journal of Linear and Topological Algebra

مقاله 3، دوره 02، شماره 03، آذر 2013، صفحه 145-151   

نویسندگان

A. A. Tabatabai Adnani^{* 1}؛ A. Reza¹؛ M. Morovati²

¹Islamic Azad University, Central Tehran Branch, Tehran, Iran

²School of Automotive Engineering, Iran University of Science and Technology, Tehran, Iran

چکیده

One of the most practical routine tests for convergence of a positive series makes use of the ratio test. If this test fails, we can use Rabbe's test. When Rabbe's test fails the next sharper criteria which may sometimes be used is the Bertrand's test. If this test fails, we can use a generalization of Bertrand's test and such tests can be continued innitely. For simplicity, we call ratio test, Rabbe's test, Bertrand's test as the Bertrand's test of order 0, 1 and 2, respectively. In this paper, we generalize Bertrand's test in order k for natural k > 2. It is also shown that for any k, there exists a series such that the Bertrand's test of order fails, but such test of order k + 1 is useful, furthermore we show that there exists a series such that for any k, Bertrand's test of order k fails. The only prerequisite for reading this article is a standard knowledge of advanced calculus.

کلیدواژه‌ها

Bertrand's test؛ Convergence test؛ Series test

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