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

, then either both series converge or both series diverge.

Proof

, or equivalently

and by the direct comparison test.

.

That is, both series converge or both series diverge.

Example

we have that the original series also converges.

One-sided version

One can state a one-sided comparison test by using limit superior converges.

Example

converges.

Converse of the one-sided comparison test

.

Example

and have image of finite area. By Parseval's formula.

See also

• Convergence tests

• Direct comparison test

References

Further reading

• Rinaldo B. Schinazi: From Calculus to Analysis. Springer, 2011, , 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