Limit comparison test

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In mathematics, the limit comparison test (LCT) (in contrast with the related direct comparison test) is a method of testing for the convergence of an infinite series.

Statement[edit]

Suppose that we have two series and with for all .

Then if with then either both series converge or both series diverge.

Proof[edit]

Because we know that for all there is an 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 converges, again by the direct comparison test, so does .

That is both series converge or both series diverge.

Example[edit]

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[edit]

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

Example[edit]

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[edit]

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[edit]

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, .

See also[edit]

External links[edit]