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Root Test

Let $u_k$ be a Series with Positive terms, and let

\rho\equiv \lim_{k\to \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.

See also Convergence Tests


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.

© 1996-9 Eric W. Weisstein