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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.


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


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.


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[edit]

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


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 .


Let be analytic in the unit disc and have image of finite area. By Parseval's formula the area of the image of is proportional to . Moreover, diverges. Therefore, by the converse of the comparison test, we have , that is, .

See also[edit]


  1. ^ Swokowski, Earl (1983), Calculus with analytic geometry (Alternate ed.), Prindle, Weber & Schmidt, p. 516, ISBN 0-87150-341-7

Further reading[edit]

  • 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[edit]