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Beyond the ratio test

Published online by Cambridge University Press:  17 October 2018

G. J. O. Jameson
G. J. O. Jameson*
Affiliation:
Dept. of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF e-mail: g.jameson@lancaster.ac.uk

Extract

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D'Alembert's ratio test, a very basic plank in the theory of infinite series, can be stated as follows:

Suppose that an > 0 for all n ≥ 1. Then:

  1. (i) if for some n0 and some ρ < 1, we have for all n ≥ n0, then is convergent;

  2. (ii) if for some n0, we have for all n ≥ n0, then is divergent.

Information

Type
Articles
Information
The Mathematical Gazette , Volume 102 , Issue 555 , November 2018, pp. 471 - 484
Copyright
Copyright © Mathematical Association 2018 

References

1. Hyslop, J. M., Infinite series, Oliver and Boyd (1959).Google Scholar
2. Phillips, E. G., A course of analysis (2nd edn.), Cambridge (1939).Google Scholar
3. Ferrar, W. L., A textbook of convergence, Oxford University Press (1938).Google Scholar
4. Jameson, G. J. O., A fresh look at Euler's limit formula for the gamma function, Math. Gaz. 98 (July 2014) pp. 235–242.Google Scholar
5. Scott, J. A., On series for the Euler constant, Math. Gaz. 101 (November 2017) pp. 486–488.Google Scholar