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Root Test

Let sum_(k=1)^(infty)u_k be a series with positive terms, and let

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

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.

This test is also called the Cauchy root test (Zwillinger 1996, p. 32).

See also

Convergence Tests

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 281-282, 1985.Bromwich, T. J. I'A. and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, pp. 31-39, 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.

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Root Test

Cite this as:

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

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