This file is part of a distribution of the Calculus Applets website (http://www.calculusapplets.com) (v1.1) which has been reformatted for the needs of this OpenCourseware course.

The **integral test** provides another means to testing whether a series converges or diverges. Suppose we have a sequence defined by *a _{n}* =

Try the following:

- The applet shows the harmonic series. Note that the graph also shows a plot of
*f*(*x*) = 1/*x*as a blue line. Since this is positive, decreasing and continuous, we can use the integral test. The integral can be evaluated by. Since ln*x*grows without bound, the last limit does not exist, so the harmonic series diverges.

- Select the second example, where the series is . From looking at the table and the graph, it isn't quite clear whether this converges or not. The blue line becomes positive and decreasing for
*x*> 1, so we can use the integral test: , where we used the substitution*u*=*x*² + 1. The limit clearly doesn't exist, so this series diverges.

- Select the third example, showing the series . From the graph and table it looks like this series does converge, but we can verify this with the integral test. Since e
is simple to integrate and is positive, decreasing, and continuous for all^{-x}*x*, we can use the integral test: . Since this limit is zero, due to the minus sign in the exponent, the series converges. Note that we used a lower limit of 0 here, instead of 1, just to make the evaluation of the integral a little bit easier.

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© Copyright 2001 David J. Eck

© Copyright 2007 Thomas S. Downey

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Citation: tdowney. (2009, August 06). Integral Test. Retrieved September 23, 2014, from Notre Dame OpenCourseWare Web site: http://ocw.nd.edu/mathematics/calculus-ii-for-business/calculus-applets-website/Integral%20Test.html.

Copyright 2012,
Thomas S. Downey.
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