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| {{Calculus |Series}}
| | Andrew Berryhill is what his wife enjoys to call him and he completely digs that title. One of the issues she enjoys most is canoeing and she's been doing it for fairly a while. I've always cherished residing in Kentucky but now I'm contemplating other options. I am an invoicing officer and I'll be promoted soon.<br><br>Feel free to visit my weblog :: psychic phone readings ([http://appin.co.kr/board_YwqL15/221846 appin.co.kr]) |
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| 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 the convergence or divergence of an [[series (mathematics)|infinite series]] or an [[improper integral]]. In both cases, the test works by comparing the given series or integral to one whose convergence properties are known.
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| == For series ==
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| In [[calculus]], the comparison test for series typically consists of a pair of statements about infinite series with nonnegative ([[Real number|real-valued]]) terms:<ref>Ayres & Mendelson (1999), p. 401.</ref>
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| * 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.
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| * If the infinite series <math>\sum b_n</math> diverges and <math>a_n \ge b_n \ge 0</math> for all sufficiently large ''n'', then the infinite series <math>\sum a_n</math> also diverges.
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| Note that the series having larger terms is sometimes said to ''dominate'' (or ''eventually dominate'') the series with smaller terms.<ref>Munem & Foulis (1984), p. 662.</ref>
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| 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>
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| * 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.
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| * If the infinite series <math>\sum b_n</math> is not absolutely convergent and <math>|a_n| \ge |b_n|</math> for all sufficiently large ''n'', then the infinite series <math>\sum a_n</math> is also not absolutely convergent.
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| 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<sub>n</sub>'' are not all nonnegative.
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| 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.
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| ===Proof===
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| The proofs of all the statements given above are similar. Here is a proof of the third statement.
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| 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
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| :<math>S_n = |a_1| + |a_2| + \ldots + |a_n|,\ T_n = |b_1| + |b_2| + \ldots + |b_n|. </math>
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| Since <math>\sum b_n</math> converges absolutely, <math>\lim_{n\to\infty} T_n = T</math> for some real number ''T''. The sequence <math>T_n</math> is clearly nondecreasing, so <math>T_n \le T</math> for all ''n''. Thus for all ''n'',
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| :<math> 0 \le S_n = |a_1| + |a_2| + \ldots + |a_n| \le |b_1| + |b_2| + \ldots + |b_n| \le T.</math> | |
| This shows that <math>S_n</math> is a bounded [[monotonic sequence]] and so must converge to a limit. Therefore <math>\sum a_n</math> is absolutely convergent.
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| ==For integrals==
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| 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>
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| * 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>
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| * 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.
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| ==Ratio comparison test==
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| 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>
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| * 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.
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| ==Notes==
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| <references />
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| ==References==
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| * {{cite book|last1=Ayres|first1=Frank Jr.|last2=Mendelson|first2=Elliott|authorlink2=Elliott Mendelson|title=Schaum's Outline of Calculus|edition=4th|publisher=McGraw-Hill|location=New York|date=1999|isbn=0-07-041973-6}}
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| * {{cite book|last=Buck|first=R. Creighton|authorlink=Robert Creighton Buck|title=Advanced Calculus|edition=2nd|date=1965|publisher=McGraw-Hill|location=New York}}
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| * {{cite book|last=Knopp|first=Konrad|authorlink=Konrad Knopp|title=Infinite Sequences and Series|publisher=Dover Publications|location=New York|date=1956|at=§ 3.1|isbn=0-486-60153-6}}
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| * {{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}}
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| * {{cite book|last=Silverman|first1=Herb|title=Complex Variables|date=1975|publisher=Houghton Mifflin Company|isbn=0-395-18582-3}}
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| * {{cite book|last1=Whittaker|first1=E. T.|authorlink1=E. T. Whittaker|last2=Watson|first2=G. N.|authorlink2=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}}
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| ==See also==
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| *[[Convergence tests]]
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| *[[Convergence (mathematics)]]
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| *[[Dominated convergence theorem]]
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| *[[Integral test for convergence]]
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| *[[Limit comparison test]]
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| *[[Monotone convergence theorem]]
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| [[Category:Convergence tests]]
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| [[Category:Mathematical series]]
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| [[bs:Test poređenja]]
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| [[de:Majorantenkriterium]]
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| [[fr:Série convergente#Principe général : règles de comparaison]]
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| [[ko:비교판정법]]
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| [[hu:Majoráns kritérium]]
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| [[ja:比較判定法]]
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| [[pt:Teste da comparação]]
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| [[ro:Criteriile de comparație]]
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| [[ru:Признак сравнения]]
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| [[fi:Vertailuperiaate]]
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| [[tr:Karşılaştırma testi]]
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| [[zh:比较审敛法]]
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Andrew Berryhill is what his wife enjoys to call him and he completely digs that title. One of the issues she enjoys most is canoeing and she's been doing it for fairly a while. I've always cherished residing in Kentucky but now I'm contemplating other options. I am an invoicing officer and I'll be promoted soon.
Feel free to visit my weblog :: psychic phone readings (appin.co.kr)