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 ≤ a_{n} < b_{n}. 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.
1. Close to a P series
The initial applet shows the series
This is similar to a pseries, so the applet also shows a pseries as B. The blue dots are terms of A and the blue/purple rectangles are the terms of the underlying sequence a_{n}. The red dots represent B and the red/pink rectangles are the terms b_{n}.
Note that all of the a_{n} are less than the corresponding b_{n} and that all are positive, so we can apply the comparison test. Since we know that a pseriese 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
This is similar to a harmonic series, which is shown as A. Note that all of the b_{n} are greater than the corresponding a_{n} 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.
Other 'Sequences and Series' topics
Acknowledgements
Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.
