The mth Ratio Test: New Convergence Tests for Series

@article{Ali2008TheMR,
  title={The mth Ratio Test: New Convergence Tests for Series},
  author={Sayel A. Ali},
  journal={The American Mathematical Monthly},
  year={2008},
  volume={115},
  pages={514 - 524}
}
  • Sayel A. Ali
  • Published 1 June 2008
  • Mathematics
  • The American Mathematical Monthly
The famous ratio test of d’Alembert for convergence of series depends on the limit of the simple ratio an+1 an (J. d’Alembert, 1717–1783). If the limit is 1, the test fails. Most notable is its failure in situations where it is expected to succeed. For example, it often fails on series with terms containing factorials or finite products. Such terms appear in Taylor series of many functions. The frequent failure of the ratio test motivated many mathematicians to analyze the ratio an+1 an when… 
The phi-ratio tests
In [1], a family of new convergence tests, the m-th ratio tests, was established. These tests are stronger than the ordinary ratio test; that is, they succeed in testing many series for which the
A Second Look at the Second Ratio Test
TLDR
A class of series convergence tests, known as the mth ratio tests, that were introduced by Sayel A. Ali in 2008 are examined, namely Raabe's test and Jamet's test, that are based on an implicit comparison with a p-series.
Regular Variation and Raabe
There are many tests for determining the convergence or divergence of series. The test of Raabe and the test of Betrand are relatively unknown and do not appear in most classical courses of analysis.
C A ] 6 A ug 2 01 8 Regular Variation and Raabe
There are many tests for determining the convergence or divergence of series. The test of Raabe and the test of Betrand are relatively unknown and do not appear in most classical courses of analysis.
Some open problems concerning the convergence of positive series
We discuss some old results due to Abel and Olivier concerning the convergence of positive series and prove a set of necessary conditions involving convergence in density.
The Second Raabe's Test and Other Series Tests
The classical D’Alembert’s Ratio Test is a powerful test that we learn from calculus to determine convergence for a series of positive terms. Its range of applicability and ease of computation makes
Series of Real Numbers

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so π is an “infinite sum” of fractions. Decimal expansions like this show that an infinite series is not a paradoxical idea, although it may not be clear how to deal with non-decimal infinite series
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  • New York,
  • 1967
United Arab Emirates saali@pi.ac.ae alis@mnstate
  • Mathematics