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.
Contents
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 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, .
See also
References
 ^ Swokowski, Earl (1983), Calculus with analytic geometry (Alternate ed.), Prindle, Weber & Schmidt, p. 516, ISBN 0871503417
Further reading
 Rinaldo B. Schinazi: From Calculus to Analysis. Springer, 2011, ISBN 9780817682897, 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)
External links
 Pauls Online Notes on Comparison Test