The Root Test

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

Here's an example of the root test. Look at the series

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

Find the limit of the nth root of the nth term.

the limit as n goes to infinity of the nth root of (3n)/(2^n) = the limit as n goes to infinity of (the nth root of 3n)/2 = 1/2*the limit as n goes to infinity of exp(1/n *ln(3n)) =

1/2*exp(the limit as n goes to infinity of (ln(3n))/n) *= 1/2*exp(the limit as n goes to infinity of (3/3n)/1) = 1/2*e^0 = 1/2

Note the use of l'Hôpital's Rule in determining the limit. Since this limit is less than 1, the series converges.

Copyright © 1996 Department of Mathematics, Oregon State University

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