Comparison Test

The comparison test provides a way to use the convergence of a series we know to help us determine the convergence of a new series. Suppose we have two series A(inf) and b series where 0 ≤ an < bn. Then if B converges, so does A. Also, if A diverges, then so does B. So if we suspect that a series A converges, we can try to find a similar series B where the terms are all bigger than the terms of A and where B is known to converge, thus proving that A converges.

Conversely, if we have a series B that we suspect diverges, we can try to find a similar series A where the terms are all smaller than the terms of B and where A is known to diverge, thus proving that B diverges.

Substitute image

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

1. Close to a P series

The initial applet shows the series conv. series This is similar to a p-series, so the applet also shows a p-series as B. The blue dots are terms of A and the blue/purple rectangles are the terms of the underlying sequence an. The red dots represent B and the red/pink rectangles are the terms bn.

Note that all of the an are less than the corresponding bn and that all are positive, so we can apply the comparison test. Since we know that a p-seriese with p > 1 converges, B converges, and hence so does A. The table on the left shows terms of A and B and supports the convergence of both series.

2. Close to a harmonic series

Select the second example from the drop down menu, showing the series almost a harmonic series This is similar to a harmonic series, which is shown as A. Note that all of the bn are greater than the corresponding an and that all are positive, so we can apply the comparison test. Since we know that the harmonic series diverges, then so must B. The table of values isn't quite clear on whether B converges or diverges, so the comparison test is useful here to determine what happens to B in the long run.

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.