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

A(inf)    and     b series
where an >0 and bn > 0.
If     limit comparison
(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.

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See About the calculus applets for operating instructions.

1. Compare to a harmonic series

The applet shows the series 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 n/n^2 = 1/n The table shows the ratio an/bn, which does seem to converge to 1. We can verify this: limit 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. Compare to a geometric series

Select the second example from the drop down menu, showing series Use the same guidelines as before, but include the exponential term also: n/n*2^n = 1/2^n 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: limit The limit comparison test says that in this case, both converge or both diverge. Since 1/2^n = (1/2)^n the B series is a geometric series with r = 1/2, which we know converges, so A also converges.

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Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.