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.

Statement[edit]

Suppose that we have two series and with for all .

Then if with then either both series converge or both series diverge.[1]

Proof[edit]

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[edit]

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[edit]

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

Example[edit]

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 .

Example[edit]

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[edit]

References[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]