Absolute convergence
In mathematics, 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. More precisely, a real or complex series formula_1 is said to converge absolutely if formula_2 for some real number formula_3 Similarly, an improper integral of a function, formula_4 is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if formula_5 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". For instance, rearrangements do not change the value of the sum. This is not true for conditionally convergent series: The alternating harmonic series formula_6 converges to formula_7 while its rearrangement formula_8 (in which the repeating pattern of signs is two positive terms followed by one negative term) converges to formula_9 Background. In finite sums, the order in which terms are added does not matter. 1 + 2 + 3 is the same as 3  + 2 + 1. However, 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 formula_10 whose terms alternate between +1 and -1. What is the value of S? One way to evaluate S is to group the first and second term, the third and fourth, and so on: formula_11 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, and so on: formula_12 This leads  to an apparent paradox: does formula_13 or formula_14? The answer is that because S is not absolutely convergent, rearranging its terms changes the value of the sum. This means formula_15 and formula_16 are not equal. In fact, the series formula_17 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. Definition for real and complex numbers. A sum of real numbers or complex numbers formula_18 is absolutely convergent if the sum of  the absolute values of the terms formula_19 converges. Sums of more general elements. The same definition can be used for series formula_18 whose terms formula_21 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 formula_22 on an abelian group formula_23 (written additively, with identity element 0) such that: In this case, the function formula_33 induces the structure of a metric space (a type of topology) on formula_34 Then, a formula_23-valued series  is absolutely convergent if formula_36 In particular, these statements apply using the norm formula_37 (absolute value) in the space of real numbers or complex numbers. In topological vector spaces. If formula_38 is a topological vector space (TVS) and formula_39 is a (possibly uncountable) family in formula_38 then this family is absolutely summable if If formula_38 is a normable space and if formula_39 is an absolutely summable family in formula_45 then necessarily all but a countable collection of formula_57's are 0. Absolutely summable families play an important role in the theory of nuclear spaces. Relation to convergence. If formula_23 is  complete with respect to the metric formula_59 then every absolutely convergent series is convergent. The proof is the same as for complex-valued series: use the completeness to derive the Cauchy criterion for convergence—a series is convergent if and only if its tails can be made arbitrarily small in norm—and apply the triangle inequality. In particular, for series with values in any Banach space, absolute convergence implies convergence. The converse is also true: if absolute convergence implies convergence in a normed space, then the space is a Banach space. If a series is convergent but not absolutely convergent , it is called conditionally convergent. An example of a conditionally convergent series is the alternating harmonic series. Many standard tests for divergence and convergence, most notably including the ratio test and the root test, demonstrate absolute convergence. This is because a power series is absolutely convergent on the interior of its disk of convergence. Proof that any absolutely convergent series of complex numbers is convergent. Suppose that formula_60 is convergent. Then equivalently, formula_61 is convergent, which implies that formula_62 and formula_63 converge by termwise comparison of non-negative terms. It suffices to show that the convergence of these series implies  the convergence of formula_64 and formula_65 for then, the convergence of formula_66 would follow, by the definition of the convergence of complex-valued series. The preceding discussion shows that we need only prove that convergence of formula_67 implies the convergence of formula_68 Let formula_67 be convergent. Since formula_70 we have formula_71 Since formula_72 is convergent, formula_73 is a bounded monotonic sequence of partial sums, and formula_74 must also converge. Noting that formula_75 is the difference of convergent series, we conclude that it too is a convergent series, as desired. Alternative proof using the Cauchy criterion and triangle inequality. By applying  the Cauchy criterion for the convergence of a complex series, we can also prove this fact as a simple implication of the triangle inequality. By the Cauchy criterion, formula_76 converges if and only if for any formula_77 there exists formula_78 such that formula_79 for any formula_80 But the triangle inequality implies that formula_81 so that formula_82 for any formula_83 which is exactly the Cauchy criterion for formula_84 Proof that any absolutely convergent series in a Banach space is convergent. The above result can be easily generalized to every Banach space formula_85 Let formula_86 be an absolutely convergent series in formula_87  As formula_88 is a Cauchy sequence of real numbers, for any formula_89 and large enough natural numbers formula_90 it holds: formula_91 By the triangle inequality for the norm , one immediately gets: formula_92 which means that formula_93 is a Cauchy sequence in formula_45 hence the series is convergent in formula_87 Rearrangements and unconditional convergence. Real and complex numbers. When a series of real or complex numbers is absolutely convergent, any rearrangement or reordering of that series' terms will still converge to the same value. This fact is one reason absolutely convergent series are useful: showing a series is absolutely convergent allows  terms to be paired or rearranged in convenient ways without changing the sum's value. The Riemann rearrangement theorem shows that the converse is also true: every real or complex-valued series whose terms cannot be reordered to give a different value is absolutely convergent. Series with coefficients in more general space. The term unconditional convergence is used to refer to a series where any rearrangement of its terms still converges to the same value. For any series with values in a normed abelian group formula_23, as long as formula_23 is complete, every series which converges absolutely also converges  unconditionally. Stated more formally: For series with more general coefficients, the converse is more complicated. As stated in the previous section, for real-valued and complex-valued series, unconditional convergence always implies absolute convergence. However, in the more general case of a series with values in any normed abelian group formula_23, the converse does not always hold: there can exist series which are not absolutely convergent, yet unconditionally convergent. For example, in the Banach space ℓ, one series which is unconditionally convergent but not absolutely convergent is: formula_99 where formula_100 is an orthonormal basis. A theorem of A. Dvoretzky and  C. A. Rogers asserts that every infinite-dimensional Banach space has an unconditionally convergent series that is not absolutely convergent. Proof of the theorem. For any formula_77 we can choose some formula_102 such that: formula_103 Let formula_104 Finally for any integer formula_105 let formula_106 Then formula_107 This shows that formula_108 that is: formula_109 Q.E.D. Products of series. The Cauchy product of two series converges to the product of the sums if at least one of the series converges absolutely. That is, suppose that formula_110 The Cauchy product is defined as the sum of terms formula_111 where: formula_112 If the formula_21  or formula_114 sum converges absolutely then formula_115 Absolute convergence over sets. A generalization of the absolute convergence of a series, is the absolute convergence of a sum of a function over a set. We can first consider a countable set formula_38 and a function formula_117 We will give a definition below of the sum of formula_118 over formula_45 written as formula_120 First note that because no particular enumeration (or "indexing") of formula_38 has yet been specified, the series formula_122 cannot be understood by the more basic definition of a series. In fact, for certain examples of formula_38 and formula_124  the sum of formula_118 over formula_38 may not be defined at all, since some indexing may produce a conditionally convergent series. Therefore we define formula_127 only in the case where there exists some bijection formula_128 such that formula_129 is absolutely convergent. Note that here, "absolutely convergent" uses the more basic definition, applied to an indexed series. In this case, the value of the sum of formula_118 over formula_38 is defined by formula_132 Note that because the series is absolutely convergent, then every rearrangement is identical to a different choice of bijection formula_133 Since all of these sums have the same  value, then the sum of formula_118 over formula_38 is well-defined. Even more generally we may define the sum of formula_118 over formula_38 when formula_38 is uncountable. But first we define what it means for the sum to be convergent. Let formula_38 be any set, countable or uncountable, and formula_140 a function. We say that the sum of formula_118 over formula_38 converges absolutely if formula_143 There is a theorem which states that, if the sum of formula_118 over formula_38 is absolutely convergent, then formula_118 takes non-zero values on a set that is at most countable. Therefore, the following is  a consistent definition of the sum of formula_118 over formula_38 when the sum is absolutely convergent. formula_149 Note that the final series uses the definition of a series over a countable set. Some authors define an iterated sum formula_150 to be absolutely convergent if the iterated series formula_151 This is in fact equivalent to the absolute convergence of formula_152 That is to say, if the sum of formula_118 over formula_45 formula_155 converges absolutely, as defined above, then the iterated sum formula_150 converges absolutely, and vice versa. Absolute convergence of integrals. The integral formula_157 of a real or complex-valued function  is said to converge absolutely if formula_158 One also says that formula_118 is absolutely integrable. The issue of absolute integrability is intricate and depends on whether the Riemann, Lebesgue, or Kurzweil-Henstock (gauge) integral is considered; for the Riemann integral, it also depends on whether we only consider integrability in its proper sense (formula_118 and formula_47 both bounded), or permit the more general case of improper integrals. As a standard property of the Riemann integral, when formula_162 is a bounded interval, every continuous function is bounded and (Riemann) integrable, and since formula_118 continuous implies formula_164 continuous, every continuous function is  absolutely integrable. In fact, since formula_165 is Riemann integrable on formula_166 if formula_118 is (properly) integrable and formula_168 is continuous, it follows that formula_169 is properly Riemann integrable if formula_118 is. However, this implication does not hold in the case of improper integrals. For instance, the function formula_171 is improperly Riemann integrable on its unbounded domain, but it is not absolutely integrable: formula_172 Indeed, more generally, given any series formula_173 one can consider the associated step function formula_174 defined by formula_175 Then formula_176 converges absolutely, converges conditionally or diverges according to the corresponding behavior of formula_177 The situation is different  for the Lebesgue integral, which does not handle bounded and unbounded domains of integration separately ("see below"). The fact that the integral of formula_164 is unbounded in the examples above implies that formula_118 is also not integrable in the Lebesgue sense. In fact, in the Lebesgue theory of integration, given that formula_118 is measurable, formula_118 is (Lebesgue) integrable if and only if formula_164 is (Lebesgue) integrable. However, the hypothesis that formula_118 is measurable is crucial; it is not generally true that absolutely integrable functions on formula_166 are integrable (simply because they may fail to be measurable): let formula_185 be a  nonmeasurable subset and consider formula_186 where formula_187 is the characteristic function of formula_188 Then formula_118 is not Lebesgue measurable and thus not integrable, but formula_190 is a constant function and clearly integrable. On the other hand, a function formula_118 may be Kurzweil-Henstock integrable (gauge integrable) while formula_164 is not. This includes the case of improperly Riemann integrable functions. In a general sense, on any measure space formula_193 the Lebesgue integral of a real-valued function is defined in terms of its positive and negative parts, so the facts: are essentially built into the definition of the Lebesgue integral. In  particular, applying the theory to the counting measure on a set formula_199 one recovers the notion of unordered summation of series developed by Moore–Smith using (what are now called) nets. When formula_200 is the set of natural numbers, Lebesgue integrability, unordered summability and absolute convergence all coincide. Finally, all of the above holds for integrals with values in a Banach space. The definition of a Banach-valued Riemann integral is an evident modification of the usual one. For the Lebesgue integral one needs to circumvent the decomposition into positive and negative parts with Daniell's more functional analytic approach , obtaining the Bochner integral.