Calculus Applets

Limit Comparison Test

The limit comparison test is similar to the comparison test in that you use another series to show the convergence or divergence of a desired series. Suppose we have two series and , where a_n >0 and b_n > 0. If (i.e., if the ratio of the terms tends to a finite number as n goes to infinity), then both series converge or both series diverge. By picking a suitable B, usually a p-series, we can use this test to determine whether or not A converges.

Try the following:

1. The applet shows the series . A useful way to pick a comparison series when the target series uses a rational expression is to divide the highest power of n in the numerator by the highest power of n in the denominator, which in this case yields . The table shows the ratio a_n/b_n, which does seem to converge to 1. We can verify this: . The limit comparison test says that in this case, both converge or both diverge. Since we know that the harmonic series diverges, A must also diverge.
   
   
2. Select the second example from the drop down menu, showing . Use the same guidelines as before, but include the exponential term also: . The limit of the ratio seems to converge to 1 (the "undefined" in the table is due to the b terms getting so small that the algorithm thinks it is dividing by 0), which we can verify: . The limit comparison test says that in this case, both converge or both diverge. Since , the B series is a geometric series with r = 1/2, which we know converges, so A also converges.

This work by Thomas S. Downey is licensed under a Creative Commons Attribution 3.0 License.