limit comparison test

The Limit Comparison Test
If the limit of a[n]/b[n] is positive, then the sum of a[n] converges if and only if the sum of b[n] converges. If the limit of a[n]/b[n] is zero, and the sum of b[n] converges, then the sum of a[n] also converges.


The Limit Comparison Test (examples, solutions, videos)



 this section we will discuss using the Comparison Test and Limit Comparison Tests to determine if an infinite series converges or diverges.
In mathematics, the limit comparison test (LCT) is a method of testing for the convergence of an infinite series. Contents. 1 Statement; 2 Proof; 3 Example .

Statement

Suppose that we have two series  and  with  for all .
Then if  with , then either both series converge or both series diverge.[1]

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, .


The limit comparison test shows that the original series is divergent. The limit comparison test does not apply because the limit in question does not exist. The ever useful Limit Comparison Test will save the day! using the comparison tests to determine convergence of an infinite series. Comparison Test Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Limit Comparison Test Example Example Example Example Example.

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