The Divergence Test



If the limit of a[n] is not zero, or does not exist, then the sum diverges.

For instance, the sum

the
sum over n from 1 to infinity of (n+1)/n

doesn't converge, since the limit as n goes to infinity of (n+1)/n is 1. Note that the implication only goes one way; if the limit is zero, you still may not get convergence. For instance, the terms of

the sum over
n from 1 to infinity of 1/n

have a limit of zero, but the sum does not converge.


Copyright © 1996 Department of Mathematics, Oregon State University

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