Note that if
and
diverges, the limit comparison test gives no information. Similarly, if
and
converges, the test also provides no information. For example, consider the two series
and
These series are both
p -series with
and
respectively. Since
the series
diverges. On the other hand, since
the series
converges. However, suppose we attempted to apply the limit comparison test, using the convergent
as our comparison series. First, we see that
Similarly, we see that
Therefore, if
when
converges, we do not gain any information on the convergence or divergence of
Using the limit comparison test
For each of the following series, use the limit comparison test to determine whether the series converges or diverges. If the test does not apply, say so.
Compare this series to
Calculate
By the limit comparison test, since
diverges, then
diverges.
Compare this series to
We see that
Therefore,
Since
converges, we conclude that
converges.
Since
compare with
We see that
In order to evaluate
evaluate the limit as
of the real-valued function
These two limits are equal, and making this change allows us to use L’Hôpital’s rule. We obtain
Therefore,
and, consequently,
Since the limit is
but
diverges, the limit comparison test does not provide any information.
Compare with
instead. In this case,
Since the limit is
but
converges, the test still does not provide any information.
So now we try a series between the two we already tried. Choosing the series
we see that
As above, in order to evaluate
evaluate the limit as
of the real-valued function
Using L’Hôpital’s rule,
Since the limit is
and
converges, we can conclude that
converges.