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 converges or diverges. If f is positive, decreasing and continuous for x > c, then if converges 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:
- 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 = x² + 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-x is simple to integrate and is positive, decreasing, and continuous for all 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.
This work by Thomas S. Downey is licensed under a Creative Commons Attribution 3.0 License.