The Absolute Convergence Test

If the sum of |a[n]| converges, then the sum of a[n] converges.

We call this type of convergence absolute convergence.

As an example, look at

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

We know that since the absolute value of sin(x) is always less than or equal to one, then

|(sin(n))/(n^2)| <= 1/(n^2)

So, by the Comparison Test, and the fact that

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

is a convergent p-series, we find that

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

converges, so

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

converges.

Copyright © 1996 Department of Mathematics, Oregon State University

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