Limit comparison test

Summary

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

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

Proof

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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

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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

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

Example

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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

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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

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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

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References

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

Further reading

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  • 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)
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  • Pauls Online Notes on Comparison Test