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Ratio Test

Let u_k be a series with positive terms and suppose

rho=lim_(k->infty)(u_(k+1))/(u_k).

Then

1. If rho<1, the series converges.

2. If rho>1 or rho=infty, the series diverges.

3. If rho=1, the series may converge or diverge.

The test is also called the Cauchy ratio test or d'Alembert ratio test.

See also

Convergence Tests Explore this topic in the MathWorld classroom

Explore with Wolfram|Alpha

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 282-283, 1985.Bromwich, T. J. I'A. and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, p. 28, 1991.Zwillinger, D. (Ed.). "Convergence Tests." §1.3.3 in CRC Standard Mathematical Tables and Formulae, 30th ed. Boca Raton, FL: CRC Press, p. 32, 1996.

Referenced on Wolfram|Alpha

Ratio Test

Cite this as:

Weisstein, Eric W. "Ratio Test." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RatioTest.html

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