The ratio test is helpful for determining the convergence of a series when the terms grow very large as n increases.. Suppose we have a series series where the an are positive. If lim ratio test then if N < 1 the series converges, if N > 1 the series diverges, and if N = 1 we don't know whether it diverges or converges.

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See About the calculus applets for operating instructions.

The applet shows the series factorial series From the graph and table it definitely looks like this series converges, and quite rapidly, too (the "undefined" entries in the table are due to the n! becoming so large that the value exceeds the capacity of the variable storing the number). The ratio test says that we want to look at the ratio of successive terms as n gets large: ratio test with factorial Since the limit is < 1, the series converges.

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Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.