Limit comparison test

Short description: Method of testing for the convergence of an infinite series

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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.

## Contents

* 1 Statement
* 2 Proof
* 3 Example
* 4 One-sided version
* 5 Example
* 6 Converse of the one-sided comparison test
* 7 Example
* 8 See also
* 9 References
* 10 Further reading
* 11 External links

## Statement

Suppose that we have two series Σnan and Σnbn with an≥0,bn>0 for all n. Then if limn->∞anbn=c with 0<c<∞, then either both series converge or both series diverge.[1]

## Proof

Because limn->∞anbn=c we know that for every ε>0 there is a positive integer n0 such that for all n≥n0 we have that |anbn−c|<ε, or equivalently

−ε<anbn−c<ε

c−ε<anbn<c+ε

(c−ε)bn<an<(c+ε)bn

As c>0 we can choose ε to be sufficiently small such that c−ε is positive. So bn<1c−εan and by the direct comparison test, if ∑nan converges then so does ∑nbn.

Similarly an<(c+ε)bn, so if ∑nan diverges, again by the direct comparison test, so does ∑nbn.

That is, both series converge or both series diverge.

## Example

We want to determine if the series ∑n=1∞1n2+2n converges. For this we compare it with the convergent series ∑n=1∞1n2=π26

As limn->∞1n2+2nn21=1>0 we have that the original series also converges.

## One-sided version

One can state a one-sided comparison test by using limit superior. Let an,bn≥0 for all n. Then if lim supn->∞anbn=c with 0≤c<∞ and Σnbn converges, necessarily Σnan converges.

## Example

Let an=1−(−1)nn2 and bn=1n2 for all natural numbers n. Now limn->∞anbn=limn->∞(1−(−1)n) does not exist, so we cannot apply the standard comparison test. However, lim supn->∞anbn=lim supn->∞(1−(−1)n)=2∈[0,∞) and since ∑n=1∞1n2 converges, the one-sided comparison test implies that ∑n=1∞1−(−1)nn2 converges.

## Converse of the one-sided comparison test

Let an,bn≥0 for all n. If Σnan diverges and Σnbn converges, then necessarily lim supn->∞anbn=∞, that is, lim infn->∞bnan=0. The essential content here is that in some sense the numbers an are larger than the numbers bn.

## Example

Let f(z)=∑n=0∞anzn be analytic in the unit disc D={z∈ℂ:|z|<1} and have image of finite area. By Parseval's formula the area of the image of f is proportional to ∑n=1∞n|an|2. Moreover, ∑n=1∞1/n diverges. Therefore, by the converse of the comparison test, we have lim infn->∞n|an|21/n=lim infn->∞(n|an|)2=0, that is, lim infn->∞n|an|=0.

## See also

* Convergence tests
* Direct comparison test

## References

1. ↑ Swokowski, Earl (1983), Calculus with analytic geometry (Alternate ed.), Prindle, Weber & Schmidt, p. 516, ISBN 0-87150-341-7, https://archive.org/details/calculuswithanal00swok/page/516

## 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

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Calculus

Precalculus|

* Binomial theorem
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Differential calculus|

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Integral calculus|

* Antiderivative
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Vector calculus|

* Curl
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Multivariable calculus|

* Curvature
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Series|

* Abel's test
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* Alternating series test
* Arithmetico-geometric sequence
* Binomial
* Cauchy condensation test
* Direct comparison test
* Dirichlet's test
* Euler–Maclaurin formula
* Fourier
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* Limit comparison test
* Maclaurin
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* Ratio test
* Root test
* Taylor
* Term test

Special functions and numbers|

* Bernoulli numbers
* e (mathematical constant)
* Exponential function
* Natural logarithm
* Stirling's approximation

History of calculus|

* Adequality
* Brook Taylor
* Colin Maclaurin
* Generality of algebra
* Gottfried Wilhelm Leibniz
* Infinitesimal
* Infinitesimal calculus
* Isaac Newton
* Fluxion
* Law of Continuity
* Leonhard Euler
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* The Method of Mechanical Theorems

Lists|

* Differentiation rules
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* List of integrals of logarithmic functions
* List of integrals of rational functions
* List of integrals of trigonometric functions
* List of limits
* List of mathematical symbols
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