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 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.

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Try the following:

1. 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 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 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. 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.

For more information on rights and downloading, refer to http://www.calculusapplets.com/download.html.

© Copyright 2001 David J. Eck

© Copyright 2007 Thomas S. Downey

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