Integral Test

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 an = f (n), where f is some function, and we want to know whether the series series of f(n)converges or diverges. If f is positive, decreasing and continuous for x > c, then if integralconverges the series also converges. If the integral diverges then so does the series. Hence if we can integrate f, and if there is some c for which f is positive, decreasing and continuous for x > c, then we can use this test. c = 1 is the most commonly selected c to use, but depending on the function you may have to use a larger c.

Try the following:

  1. 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 byintegral of 1/x. Since ln x grows without bound, the last limit does not exist, so the harmonic series diverges.

  2. Select the second example, where the series is series. 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: integral test, where we used the substitution u = x² + 1. The limit clearly doesn't exist, so this series diverges.

  3. Select the third example, showing the series exp series. From the graph and table it looks like this series does converge, but we can verify this with the integral test. Since e-x is simple to integrate and is positive, decreasing, and continuous for all x, we can use the integral test: int. 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 August 28, 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. This work is licensed under a Creative Commons Attribution 3.0 License. Creative Commons Attribution 3.0 License