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 *a _{n}* are positive. If
then if

See About the calculus applets for operating instructions. |

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.

- Sequences
- Series
- Integral Test
- Comparison Test
- Limit Comparison Test
- Ratio Test
- Alternating Series and Absolute Convergence
- Power Series & Interval of Convergence
- Taylor Series & Polynomials
- Lagrange Remainder

(C) 2011 Copyright Math Open Reference.

All rights reserved

All rights reserved