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{{Short description|Way of deducing the convergence or divergence of an infinite series or an improper integral}}

In [[mathematics]], the '''Comparison test''' is a criterion for convergence or divergence of a [[series (mathematics)|series]] whose terms are real or complex numbers. It determines convergence by comparing the terms of the series in question with those of a series whose convergence properties are known. There are two versions of the comparison test.

{{Calculus |Series}}

In [[mathematics]], the '''comparison test''', sometimes called the '''direct comparison test''' to distinguish it from similar related tests (especially the [[limit comparison test]]), provides a way of deducing whether an [[series (mathematics)|infinite series]] or an [[improper integral]] converges or diverges by comparing the series or integral to one whose convergence properties are known.

==Comparison test of the first kind==

== For series ==

The first comparison test states that if the series

In [[calculus]], the comparison test for series typically consists of a pair of statements about infinite series with non-negative ([[Real number|real-valued]]) terms:<ref>Ayres &amp; Mendelson (1999), p. 401.</ref>

:<math>\sum_{n=1}^\infty b_n</math>

* If the infinite series <math>\sum b_n</math> converges and <math>0 \le a_n \le b_n</math> for all sufficiently large ''n'' (that is, for all <math>n>N</math> for some fixed value ''N''), then the infinite series <math>\sum a_n</math> also converges.

* If the infinite series <math>\sum b_n</math> diverges and <math>0 \le b_n \le a_n</math> for all sufficiently large ''n'', then the infinite series <math>\sum a_n</math> also diverges.

Note that the series having larger terms is sometimes said to ''dominate'' (or ''eventually dominate'') the series with smaller terms.<ref>Munem &amp; Foulis (1984), p. 662.</ref>

Alternatively, the test may be stated in terms of [[absolute convergence]], in which case it also applies to series with [[complex number|complex]] terms:<ref>Silverman (1975), p. 119.</ref>

is an [[absolutely convergent]] series and there exists a real number ''C''&nbsp; independent of ''n''&nbsp; such that

* If the infinite series <math>\sum b_n</math> is absolutely convergent and <math>|a_n| \le |b_n|</math> for all sufficiently large ''n'', then the infinite series <math>\sum a_n</math> is also absolutely convergent.

* If the infinite series <math>\sum b_n</math> is not absolutely convergent and <math>|b_n| \le |a_n|</math> for all sufficiently large ''n'', then the infinite series <math>\sum a_n</math> is also not absolutely convergent.

Note that in this last statement, the series <math>\sum a_n</math> could still be [[Conditional convergence|conditionally convergent]]; for real-valued series, this could happen if the ''a_n'' are not all nonnegative.

The second pair of statements are equivalent to the first in the case of real-valued series because <math>\sum c_n</math> converges absolutely if and only if <math>\sum |c_n|</math>, a series with nonnegative terms, converges.

:<math>|a_n|\le C|b_n|</math>

===Proof===

for sufficiently large ''n''&nbsp;, then the series

The proofs of all the statements given above are similar. Here is a proof of the third statement.

Let <math>\sum a_n</math> and <math>\sum b_n</math> be infinite series such that <math>\sum b_n</math> converges absolutely (thus <math>\sum |b_n|</math> converges), and [[without loss of generality]] assume that <math>|a_n| \le |b_n|</math> for all positive integers ''n''. Consider the [[partial sum]]s

:<math>\sum_{n=1}^\infty a_n</math>

:<math>S_n = |a_1| + |a_2| + \ldots + |a_n|,\ T_n = |b_1| + |b_2| + \ldots + |b_n|. </math>

Since <math>\sum b_n</math> converges absolutely, <math>\lim_{n\to\infty} T_n = T</math> for some real number ''T''. For all ''n'',

:<math> 0 \le S_n = |a_1| + |a_2| + \ldots + |a_n| \le |a_1| + \ldots + |a_n| + |b_{n+1}| + \ldots = S_n + (T-T_n) \le T.</math>

<math>S_n</math> is a nondecreasing sequence and <math>S_n + (T - T_n)</math> is nonincreasing.

Given <math>m,n > N</math> then both <math>S_n, S_m</math> belong to the interval <math>[S_N, S_N + (T - T_N)]</math>, whose length <math>T - T_N</math> decreases to zero as <math>N</math> goes to infinity.

This shows that <math>(S_n)_{n=1,2,\ldots}</math> is a [[Cauchy sequence]], and so must converge to a limit. Therefore, <math>\sum a_n</math> is absolutely convergent.

==For integrals==

converges absolutely. If the series &sum;|''b_n''&nbsp;| is divergent and

The comparison test for integrals may be stated as follows, assuming [[Continuous function|continuous]] real-valued functions ''f'' and ''g'' on <math>[a,b)</math> with ''b'' either <math>+\infty</math> or a real number at which ''f'' and ''g'' each have a vertical asymptote:<ref>Buck (1965), p. 140.</ref>

* If the improper integral <math>\int_a^b g(x)\,dx</math> converges and <math>0 \le f(x) \le g(x)</math> for <math>a \le x < b</math>, then the improper integral <math>\int_a^b f(x)\,dx</math> also converges with <math>\int_a^b f(x)\,dx \le \int_a^b g(x)\,dx.</math>

* If the improper integral <math>\int_a^b g(x)\,dx</math> diverges and <math>0 \le g(x) \le f(x)</math> for <math>a \le x < b</math>, then the improper integral <math>\int_a^b f(x)\,dx</math> also diverges.

==Ratio comparison test==

:<math>|a_n|\ge |b_n|</math>

Another test for convergence of real-valued series, similar to both the direct comparison test above and the [[ratio test]], is called the '''ratio comparison test''':<ref>Buck (1965), p. 161.</ref>

* If the infinite series <math>\sum b_n</math> converges and <math>a_n>0</math>, <math>b_n>0</math>, and <math>\frac{a_{n+1}}{a_n} \le \frac{b_{n+1}}{b_n}</math> for all sufficiently large ''n'', then the infinite series <math>\sum a_n</math> also converges.

* If the infinite series <math>\sum b_n</math> diverges and <math>a_n>0</math>, <math>b_n>0</math>, and <math>\frac{a_{n+1}}{a_n} \ge \frac{b_{n+1}}{b_n}</math> for all sufficiently large ''n'', then the infinite series <math>\sum a_n</math> also diverges.

==See also==

for sufficiently large ''n''&nbsp;, then the series &sum;''a_n''&nbsp; also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the ''a_n''&nbsp; alternate in sign).

{{Portal|Mathematics}}

*[[Convergence tests]]

*[[Convergence (mathematics)]]

*[[Dominated convergence theorem]]

* [[RadiusIntegral test offor convergence]]

*[[Limit comparison test]]

*[[Monotone convergence theorem]]

==Notes==

==Comparison test of the second kind==

<references />

The second comparison test states that if the series

:<math>\sum_{n=1}^\infty b_n</math>

is an absolutely convergent series and there exists a real number ''C''&nbsp; independent of ''n''&nbsp; such that

:<math>\left|\frac{a_{n+1}}{a_n}\right|\le C\,\left|\frac{b_{n+1}}{b_n}\right|</math>

for sufficiently large ''n''&nbsp;, then the series

:<math>\sum_{n=1}^\infty a_n</math>

converges absolutely. If the series &sum;|''b_n''&nbsp;| is divergent and

:<math>\left|\frac{a_{n+1}}{a_n}\right|\ge \left|\frac{b_{n+1}}{b_n}\right|</math>

for sufficiently large ''n''&nbsp;, then the series &sum;''a_n''&nbsp; also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the ''a_n''&nbsp; alternate in sign).

This is based upon [[Jean de Rond d'Alembert]]'s [[ratio test]].

== References ==

* Knopp, Konrad, "Infinite Sequences and Series", Dover publications, Inc., New York, 1956. (&sect; 3.1) ISBN 0486601536

* Whittaker, E. T., and Watson, G. N., ''A Course in Modern Analysis'', fourth edition, Cambridge University Press, 1963. (&sect; 2.34) ISBN 0521588073

==See also==

== References ==

* [[Radius of convergence]]

* {{cite book|last1=Ayres|first1=Frank Jr.|last2=Mendelson|first2=Elliott|author-link2=Elliott Mendelson|title=Schaum's Outline of Calculus|edition=4th|publisher=McGraw-Hill|___location=New York|date=1999|isbn=0-07-041973-6|url=https://archive.org/details/schaumsoutlineof00ayre_0}}

* {{cite book|last=Buck|first=R. Creighton|author-link=Robert Creighton Buck|title=Advanced Calculus|edition=2nd|date=1965|publisher=McGraw-Hill|___location=New York}}

* {{cite book|last=Knopp|first=Konrad|author-link=Konrad Knopp|title=Infinite Sequences and Series|publisher=Dover Publications|___location=New York|date=1956|at=§ 3.1|isbn=0-486-60153-6}}

* {{cite book|last1=Munem|first1=M. A.|last2=Foulis|first2=D. J.|title=Calculus with Analytic Geometry|edition=2nd|date=1984|publisher=Worth Publishers|isbn=0-87901-236-6|url=https://archive.org/details/calculuswithanal00mune}}

* {{cite book|last=Silverman|first1=Herb|title=Complex Variables|date=1975|publisher=Houghton Mifflin Company|isbn=0-395-18582-3}}

* {{cite book|last1=Whittaker|first1=E. T.|author-link1=E. T. Whittaker|last2=Watson|first2=G. N.|author-link2=G. N. Watson|title=[[Whittaker and Watson|A Course in Modern Analysis]]|edition=4th|publisher=Cambridge University Press|date=1963|at=§ 2.34|isbn=0-521-58807-3}}

{{Calculus topics}}

[[Category:Mathematical series]]

[[Category:MathematicalConvergence seriestests]]

[[fr:Série convergente#Principe général : règles de comparaison]]

[[de:Majorantenkriterium]]