Limit comparison test
Part of a series of articles about  
Calculus  



Specialized 

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
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 converges, 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.
Onesided version
One can state a onesided 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 onesided comparison test implies that converges.
Converse of the onesided 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, .