# Absolute convergence

Absolute convergence (Redirected from Absolute convergence theorem) Jump to navigation Jump to search This article includes a list of general references, mais il manque suffisamment de citations en ligne correspondantes. Merci d'aider à améliorer cet article en introduisant des citations plus précises. (Février 2013) (Découvrez comment et quand supprimer ce modèle de message) En mathématiques, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. Plus précisément, a real or complex series {displaystyle textstyle sum _{n=0}^{infime }un_{n}} is said to converge absolutely if {displaystyle textstyle sum _{n=0}^{infime }la gauche|un_{n}droit|=L} for some real number {displaystyle textstyle L.} De la même manière, an improper integral of a function, {displaystyle textstyle int _{0}^{infime }F(X),dx,} is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, si {displaystyle textstyle int _{0}^{infime }|F(X)|dx=L.} Absolute convergence is important for the study of infinite series because its definition is strong enough to have properties of finite sums that not all convergent series possess - a convergent series that is not absolutely convergent is called conditionally convergent, while absolutely convergent series behave "nicely". Par exemple, rearrangements do not change the value of the sum. This is not true for conditionally convergent series: The alternating harmonic series {style de texte 1-{frac {1}{2}}+{frac {1}{3}}-{frac {1}{4}}+{frac {1}{5}}-{frac {1}{6}}+cdots } converge vers {style d'affichage ln 2,} while its rearrangement {style de texte 1+{frac {1}{3}}-{frac {1}{2}}+{frac {1}{5}}+{frac {1}{7}}-{frac {1}{4}}+cdots } (in which the repeating pattern of signs is two positive terms followed by one negative term) converge vers {style de texte {frac {3}{2}}dans 2.} Contenu 1 Arrière plan 1.1 Explication 2 Definition for real and complex numbers 3 Sums of more general elements 3.1 In topological vector spaces 4 Relation to convergence 4.1 Proof that any absolutely convergent series of complex numbers is convergent 4.1.1 Alternative proof using the Cauchy criterion and triangle inequality 4.2 Proof that any absolutely convergent series in a Banach space is convergent 5 Rearrangements and unconditional convergence 5.1 Real and complex numbers 5.2 Series with coefficients in more general space 5.3 Preuve du théorème 6 Products of series 7 Absolute convergence over sets 8 Absolute convergence of integrals 9 Voir également 10 Remarques 11 Références 11.1 Works cited 11.2 General references Background In finite sums, the order in which terms are added does not matter. 1 + 2 + 3 is the same as 3 + 2 + 1. Cependant, this is not true when adding infinitely many numbers, and wrongly assuming that it is true can lead to apparent paradoxes. One classic example is the alternating sum {displaystyle S=1-1+1-1+1-1...} whose terms alternate between +1 et -1. What is the value of S? One way to evaluate S is to group the first and second term, the third and fourth, etc: {style d'affichage S_{1}=(1-1)+(1-1)+(1-1)....=0+0+0...=0} But another way to evaluate S is to leave the first term alone and group the second and third term, then the fourth and fifth term, etc: {style d'affichage S_{2}=1+(-1+1)+(-1+1)+(-1+1)....=1+0+0+0...=1} This leads to an apparent paradox: does {displaystyle S=0} ou {displaystyle S=1} ?

The answer is that because S is not absolutely convergent, rearranging its terms changes the value of the sum. This means {style d'affichage S_{1}} et {style d'affichage S_{2}} are not equal. En réalité, the series {style d'affichage 1-1+1-1+...} does not converge, so S does not have a value to find in the first place. A series that is absolutely convergent does not have this problem: rearranging its terms does not change the value of the sum.

Explanation This is an example of a mathematical sleight of hand. If the terms of S are rearranged in such a way that every term remains in its original position, one finds that S is either the infinite series {displaystyle S=1-1+1-1+...+1-1+1-1} or with equal possibility, ce {displaystyle S=1-1+1-1+...+1-1+1} Evaluating S as before, by grouping every -1 with the +1 preceding it or by grouping every +1 except the first with the -1 preceding it, gives in the first case: {style d'affichage S_{1}=(1-1)+....+(1-1)=0+....+0=0} {style d'affichage S_{2}=1+(-1+1)+....+(-1+1)-1=1+0+....+0-1=1-1=0} et dans le second cas: {style d'affichage S_{1}=(1-1)+....+(1-1)+1=0+....+0+1=1} {style d'affichage S_{2}=1+(-1+1)+....+(-1+1)=1+0+...+0=1} This reveals the trick: the definition of S was interpreted as defining its last term as negative when evaluating {style d'affichage S_{1}=0} but positive when evaluating {style d'affichage S_{2}=1} when in fact the definition of S didn't define (and the rearrangement was independent of) either option.

Definition for real and complex numbers A sum of real numbers or complex numbers {textstyle sum _{n=0}^{infime }un_{n}} is absolutely convergent if the sum of the absolute values of the terms {textstyle sum _{n=0}^{infime }|un_{n}|} converge.

Sums of more general elements The same definition can be used for series {textstyle sum _{n=0}^{infime }un_{n}} whose terms {style d'affichage a_{n}} are not numbers but rather elements of an arbitrary abelian topological group. Dans ce cas, instead of using the absolute value, the definition requires the group to have a norm, which is a positive real-valued function {style de texte |cdot |:Gto mathbb {R} _{+}} on an abelian group {style d'affichage G} (written additively, with identity element 0) tel que: The norm of the identity element of {style d'affichage G} est zéro: {style d'affichage |0|=0.} Pour chaque {displaystyle xin G,} {style d'affichage |X|=0} implique {displaystyle x=0.} Pour chaque {displaystyle xin G,} {style d'affichage |-X|=|X|.} Pour chaque {style d'affichage x,yin G,} {style d'affichage |x+y|leq |X|+|y|.} Dans ce cas, the function {displaystyle d(X,y)=|x-y|} induces the structure of a metric space (a type of topology) sur {displaystyle G.} Alors, un {style d'affichage G} -valued series is absolutely convergent if {textstyle sum _{n=0}^{infime }|un_{n}|0,} there exists {displaystyle N} such {textstyle left|sum _{i=m}^{n}left|a_{i}right|right| =sum _{i=m}^{n}|a_{i}|mgeq N.} But {textstyle {big |}sum _{i=m}^{n}a_{i}{big |}leq _{i=m}^{n}|a_{i}|,} so {textstyle left|sum _{i=m}^{n}a_{i}right|mgeq N,} exactly {textstyle sum a_{i}.} above result easily generalized {displaystyle (X,|,cdot ,|).} {textstyle sum x_{n}} {displaystyle X.} As {textstyle sum _{k=1}^{n}|x_{k}|} numbers, {displaystyle varepsilon >0} large enough natural {displaystyle m>n} holds: {displaystyle left|sum _{k=1}^{m}|x_{k}|-sum _{k=1}^{n}|x_{k}|right| =sum _{k=n+1}^{m}|x_{k}|kappa _{varepsilon }&quad sum _{n=N}^{infime }|un_{n}|<{tfrac {varepsilon }{2}}\{text{ for all }}N>lambda _{varepsilon }&quad left|somme _{n=1}^{N}un_{n}-Aright|<{tfrac {varepsilon }{2}}end{aligned}}} Let {displaystyle {begin{aligned}N_{varepsilon }&=max left{kappa _{varepsilon },lambda _{varepsilon }right}\M_{sigma ,varepsilon }&=max left{sigma ^{-1}left(left{1,ldots ,N_{varepsilon }right}right)right}end{aligned}}} where {displaystyle sigma ^{-1}left(left{1,ldots ,N_{varepsilon }right}right)=left{sigma ^{-1}(1),ldots ,sigma ^{-1}left(N_{varepsilon }right)right}} so that {displaystyle M_{sigma ,varepsilon }} is the smallest natural number such that the list {displaystyle a_{sigma (0)},ldots ,a_{sigma left(M_{sigma ,varepsilon }right)}} includes all of the terms {displaystyle a_{0},ldots ,a_{N_{varepsilon }}} (and possibly others). Finally for any integer {displaystyle N>M_{sigma ,varepsilon }} laisser {style d'affichage {commencer{aligné}JE_{sigma ,varepsilon }&=left{1,ldots ,Nright}setminus sigma ^{-1}la gauche(la gauche{1,ldots ,N_{varepsilon }droit}droit)\S_{sigma ,varepsilon }&=min sigma left(JE_{sigma ,varepsilon }droit)=min left{sigma (k) : kin I_{sigma ,varepsilon }droit}\L_{sigma ,varepsilon }&=max sigma left(JE_{sigma ,varepsilon }droit)=max left{sigma (k) : kin I_{sigma ,varepsilon }droit}\fin{aligné}}} pour que {style d'affichage {commencer{aligné}la gauche|somme _{iin I_{sigma ,varepsilon }}un_{sigma (je)}droit|&leq sum _{iin I_{sigma ,varepsilon }}la gauche|un_{sigma (je)}droit|\&leq sum _{j=S_{sigma ,varepsilon }}^{L_{sigma ,varepsilon }}la gauche|un_{j}droit|&&{texte{ puisque }}JE_{sigma ,varepsilon }subseteq left{S_{sigma ,varepsilon },S_{sigma ,varepsilon }+1,ldots ,L_{sigma ,varepsilon }droit}\&leq sum _{j=N_{varepsilon }+1}^{infime }la gauche|un_{j}droit|&&{texte{ puisque }}S_{sigma ,varepsilon }geq N_{varepsilon }+1\&<{frac {varepsilon }{2}}end{aligned}}} and thus {displaystyle {begin{aligned}left|sum _{i=1}^{N}a_{sigma (i)}-Aright|&=left|sum _{iin sigma ^{-1}left({1,dots ,N_{varepsilon }}right)}a_{sigma (i)}-A+sum _{iin I_{sigma ,varepsilon }}a_{sigma (i)}right|\&leq left|sum _{j=1}^{N_{varepsilon }}a_{j}-Aright|+left|sum _{iin I_{sigma ,varepsilon }}a_{sigma (i)}right|\&0,{texte{ il existe }}M_{sigma ,varepsilon },{texte{ pour tous }}N>M_{sigma ,varepsilon }quad left|somme _{je=1}^{N}un_{sigma (je)}-Aright|

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