The Ratio Test

If the limit of |a[n+1]/a[n]| is less than 1, then the series (absolutely) converges. If the limit is larger than one, or infinite, then the series diverges.

Let's look at an example of this:

the sum over n from 1 to infinity of (3n)/(2^n)

Look at the ratio of consecutive terms, and find the limit.

limit as n goes to infin. of |(3(n+1)/(2^(n+1)))/((3n)/(2^n))|=limit as n goes to infin. of (3(n+1)*2^n)/(3n*2^(n+1))=limit as n goes to infin. of (n+1)/(2n)*=limit as n goes to infin. of 1/2 = 1/2

Note the use of l'Hôpital's Rule in the second-to-last step. This limit, being less than 1, tells us that the series converges.

Copyright © 1996 Department of Mathematics, Oregon State University

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