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 pseries, we can use this test to determine whether or not A converges.
1. Compare to a harmonic series
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. Compare to a geometric series
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.
While you are here..
... I have a small favor to ask. Over the years we have used advertising to support the site so it can remain free for everyone.
However, advertising revenue is falling and I have always hated the ads. So, would you go to Patreon and become a patron of the site?
When we reach the goal I will remove all advertising from the site.
It only takes a minute and any amount would be greatly appreciated.
Thank you for considering it! – John Page
Become a patron of the site at patreon.com/mathopenref
Other 'Sequences and Series' topics
Acknowledgements
Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.
