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
where the an are positive. If
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.
The applet shows the 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:
Since the limit is < 1, the series converges.
Other 'Sequences and Series' topics
Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.