Authors

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

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

Abstract

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.

Keywords

Main Subjects

[1] J. M. H. Olmsted, Advanced Calculus, Prentice Hall. (1961).
[2] J. Wen, T. Han, C. Gao, Convergence tests on constant Dirichlet series, Computers and Mathematics with Applications. 62 (2011) 3472-3489. 
[3] J. S. Chen, C. W. Liu, C. M. Liao, Two-dimensional Laplace-transformed power series solution for solute transport in a radially convergent flow field, Advances in Water Resources. 26 (2003) 1113-1124.
[4] P. Wonzy, Efficient algorithm for summation of some slowly convergent series, Applied Numerical Mathematics. 60 (2010) 1442-1453.
[5] E. Liflyand, S. Tikhonov, M. Zeltser, Extending tests for convergence of number series, Journal of Mathematical Analysis and Applications. 377 (2011) 194-206.
[6] A. Bartoszewicz, S. Glab, T. Poreda, On algebrability of nonabsolutely convergent series, Linear Algebra and its Applications. 435 (2011) 1025-1028.
[7] F. Moricz, A Quantitative Version of the Dirichlet-Jordan Test for Double Fourier Series, Journal of Approximation Theory. 71(1992) 344-358.