# 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} }

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…

## 8 Citations

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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…

## References

SHOWING 1-3 OF 3 REFERENCES

INFINITE SERIES

- Mathematics
- 2002

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…

Infinite Series, Macmillan

- New York,
- 1967

United Arab Emirates saali@pi.ac.ae alis@mnstate

- Mathematics