The Alternating Series Test

If a[n]=(-1)^(n+1)b[n], where b[n] is positive, decreasing, and converging to zero, then the sum of a[n] converges.

With the Alternating Series Test, all we need to know to determine convergence of the series is whether the limit of b[n] is zero as n goes to infinity.

So, given the series

sum over n from 1 to infinity of (-1)^(n+1) * 1/n

look at the limit of the non-alternating part:

limit as n goes to infinity of 1/n = 0

So, this series converges. Note that the other test dealing with negative numbers, the Absolute Convergence Test, would not tell us that this series converges.

Copyright © 1996 Department of Mathematics, Oregon State University

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