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 and , 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.
Try the following:
- The applet shows the 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.
- Select the second example from the drop down menu, showing the 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.
This work by Thomas S. Downey is licensed under a Creative Commons Attribution 3.0 License.