Ratio Test

Let $u_k$ be a Series with Positive terms and suppose

$$\rho\equiv\lim_{k\to \infty} {u_{k+1}\over u_k}.$$

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

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.