Integral Test
The integral test provides a 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 integral 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.

Substitute image

This device cannot display Java animations. The above is a substitute static image
See About the calculus applets for operating instructions.

1. Harmonic series

The initial applet shows a 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 integral of 1/x Since ln x grows without bound, the last limit does not exist, so the harmonic series diverges.

2. Divergent series

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. Convergent series

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.

While you are here..

... I have a small favor to ask. Over the years we have used advertising to support the site so it can remain free for everyone. However, advertising revenue is falling and I have always hated the ads. So, would you go to Patreon and become a patron of the site? When we reach the goal I will remove all advertising from the site.

It only takes a minute and any amount would be greatly appreciated. Thank you for considering it!   – John Page

Become a patron of the site at

Other 'Sequences and Series' topics


Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.