# 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…
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