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.
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 every 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 it 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,
.
Rinaldo B. Schinazi: From Calculus to Analysis. Springer, 2011, ISBN9780817682897, pp. 50
Michele Longo and Vincenzo Valori: The Comparison Test: Not Just for Nonnegative Series. Mathematics Magazine, Vol. 79, No. 3 (Jun., 2006), pp. 205–210 (JSTOR)
J. Marshall Ash: The Limit Comparison Test Needs Positivity. Mathematics Magazine, Vol. 85, No. 5 (December 2012), pp. 374–375 (JSTOR)