# Absolute convergence The answer is that because S is not absolutely convergent, rearranging its terms changes the value of the sum. This means {displaystyle S_{1}} and {displaystyle S_{2}} are not equal. In fact, the series {displaystyle 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, that {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: {displaystyle S_{1}=(1-1)+....+(1-1)=0+....+0=0} {displaystyle S_{2}=1+(-1+1)+....+(-1+1)-1=1+0+....+0-1=1-1=0} and in the second case: {displaystyle S_{1}=(1-1)+....+(1-1)+1=0+....+0+1=1} {displaystyle 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 {displaystyle S_{1}=0} but positive when evaluating {displaystyle 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}^{infty }a_{n}} is absolutely convergent if the sum of the absolute values of the terms {textstyle sum _{n=0}^{infty }|a_{n}|} converges.

Sums of more general elements The same definition can be used for series {textstyle sum _{n=0}^{infty }a_{n}} whose terms {displaystyle a_{n}} are not numbers but rather elements of an arbitrary abelian topological group. In that case, instead of using the absolute value, the definition requires the group to have a norm, which is a positive real-valued function {textstyle |cdot |:Gto mathbb {R} _{+}} on an abelian group {displaystyle G} (written additively, with identity element 0) such that: The norm of the identity element of {displaystyle G} is zero: {displaystyle |0|=0.} For every {displaystyle xin G,} {displaystyle |x|=0} implies {displaystyle x=0.} For every {displaystyle xin G,} {displaystyle |-x|=|x|.} For every {displaystyle x,yin G,} {displaystyle |x+y|leq |x|+|y|.} In this case, the function {displaystyle d(x,y)=|x-y|} induces the structure of a metric space (a type of topology) on {displaystyle G.} Then, a {displaystyle G} -valued series is absolutely convergent if {textstyle sum _{n=0}^{infty }|a_{n}|0,} there exists {displaystyle N} such that {textstyle left|sum _{i=m}^{n}left|a_{i}right|right|=sum _{i=m}^{n}|a_{i}|mgeq N.} But the triangle inequality implies that {textstyle {big |}sum _{i=m}^{n}a_{i}{big |}leq sum _{i=m}^{n}|a_{i}|,} so that {textstyle left|sum _{i=m}^{n}a_{i}right|mgeq N,} which is exactly the Cauchy criterion for {textstyle sum a_{i}.} Proof that any absolutely convergent series in a Banach space is convergent The above result can be easily generalized to every Banach space {displaystyle (X,|,cdot ,|).} Let {textstyle sum x_{n}} be an absolutely convergent series in {displaystyle X.} As {textstyle sum _{k=1}^{n}|x_{k}|} is a Cauchy sequence of real numbers, for any {displaystyle varepsilon >0} and large enough natural numbers {displaystyle m>n} it holds: {displaystyle left|sum _{k=1}^{m}|x_{k}|-sum _{k=1}^{n}|x_{k}|right|=sum _{k=n+1}^{m}|x_{k}|0,} we can choose some {displaystyle kappa _{varepsilon },lambda _{varepsilon }in mathbb {N} ,} such that: {displaystyle {begin{aligned}{text{ for all }}N>kappa _{varepsilon }&quad sum _{n=N}^{infty }|a_{n}|<{tfrac {varepsilon }{2}}\{text{ for all }}N>lambda _{varepsilon }&quad left|sum _{n=1}^{N}a_{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 }} let {displaystyle {begin{aligned}I_{sigma ,varepsilon }&=left{1,ldots ,Nright}setminus sigma ^{-1}left(left{1,ldots ,N_{varepsilon }right}right)\S_{sigma ,varepsilon }&=min sigma left(I_{sigma ,varepsilon }right)=min left{sigma (k) : kin I_{sigma ,varepsilon }right}\L_{sigma ,varepsilon }&=max sigma left(I_{sigma ,varepsilon }right)=max left{sigma (k) : kin I_{sigma ,varepsilon }right}\end{aligned}}} so that {displaystyle {begin{aligned}left|sum _{iin I_{sigma ,varepsilon }}a_{sigma (i)}right|&leq sum _{iin I_{sigma ,varepsilon }}left|a_{sigma (i)}right|\&leq sum _{j=S_{sigma ,varepsilon }}^{L_{sigma ,varepsilon }}left|a_{j}right|&&{text{ since }}I_{sigma ,varepsilon }subseteq left{S_{sigma ,varepsilon },S_{sigma ,varepsilon }+1,ldots ,L_{sigma ,varepsilon }right}\&leq sum _{j=N_{varepsilon }+1}^{infty }left|a_{j}right|&&{text{ since }}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,{text{ there exists }}M_{sigma ,varepsilon },{text{ for all }}N>M_{sigma ,varepsilon }quad left|sum _{i=1}^{N}a_{sigma (i)}-Aright|

Si quieres conocer otros artículos parecidos a Absolute convergence puedes visitar la categoría Integral calculus.

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