Limit comparison test


In mathematics, the limit comparison test is a method of testing for the convergence of an infinite series.

Statement

Suppose that we have two series and with for all.
Then if with, then either both series converge or both series diverge.

Proof

Because we know that for all there is a positive integer such that for all we have that, or equivalently
As we can choose to be sufficiently small such that is positive.
So and by the direct comparison test, if converges then so does.
Similarly, so if diverges, again by the direct comparison test, so does.
That is, both series converge or both series diverge.

Example

We want to determine if the series converges. For this we compare with the convergent series.
As we have that the original series also converges.

One-sided version

One can state a one-sided comparison test by using limit superior. Let for all. Then if with and converges, necessarily converges.

Example

Let and for all natural numbers. Now
does not exist, so we cannot apply the standard comparison test. However,
and since converges, the one-sided comparison test implies that converges.

Converse of the one-sided comparison test

Let for all. If diverges and converges, then necessarily
, that is,
. The essential content here is that in some sense the numbers are larger than the numbers.

Example

Let be analytic in the unit disc and have image of finite area. By Parseval's formula the area of the image of is. Moreover,
diverges. Therefore, by the converse of the comparison test, we have
, that is,