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Absolute convergence

From Wikipedia, the free encyclopedia

Mode of convergence of an infinite series

Inmathematics, aninfinite series of numbers is said toconverge absolutely (or to beabsolutely convergent) if the sum of theabsolute values of the summands is finite. More precisely, areal orcomplex series $\textstyle \sum _{n=0}^{\infty }a_{n}$ is said toconverge absolutely if $\textstyle \sum _{n=0}^{\infty }\left|a_{n}\right|=L$ for some real number $\textstyle L.$ Similarly, animproper integral of afunction, $\textstyle \int _{0}^{\infty }f(x)\,dx,$ is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if $\textstyle \int _{0}^{\infty }|f(x)|dx=L.$ A convergent series that is not absolutely convergent is calledconditionally convergent.

Absolute convergence is important for the study of infinite series, because its definition guarantees that a series will have some "nice" behaviors of finite sums that not all convergent series possess. For instance, rearrangements do not change the value of the sum, which is not necessarily true for conditionally convergent series.

Background

When adding a finite number of terms,addition is bothassociative andcommutative, meaning that grouping and rearrangement do not alter the final sum. For instance, $(1+2)+3$ is equal to both $1+(2+3)$ and $(3+2)+1$ . However, associativity and commutativity do not necessarily hold for infinite sums. One example is thealternating harmonic series

$S=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}=1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-{\frac {1}{6}}+\cdots$

whose terms are fractions that alternate in sign. This series isconvergent and can be evaluated using theMaclaurin series for the function $\ln(1+x)$ , which converges for all $x {\displaystyle x}$ satisfying $-1<x\leq 1$ :

$\ln(1+x)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}x^{n}}{n}}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+\cdots$

Substituting $x=1$ reveals that the original sum is equal to $\ln 2$ . The sum can also be rearranged as follows:

$S=\left(1-{\frac {1}{2}}\right)-{\frac {1}{4}}+\left({\frac {1}{3}}-{\frac {1}{6}}\right)-{\frac {1}{8}}+\left({\frac {1}{5}}-{\frac {1}{10}}\right)-{\frac {1}{12}}+\cdots$

In this rearrangement, thereciprocal of eachodd number is grouped with the reciprocal of twice its value, while the reciprocals of every multiple of 4 are evaluated separately. However, evaluating the terms inside the parentheses yields

$S={\frac {1}{2}}-{\frac {1}{4}}+{\frac {1}{6}}-{\frac {1}{8}}+{\frac {1}{10}}-{\frac {1}{12}}+\cdots$

or half the original series. The violation of the associativity and commutativity of addition reveals that the alternating harmonic series isconditionally convergent. Indeed, the sum of the absolute values of each term is ${\textstyle 1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+\cdots }$ , or the divergentharmonic series. According to theRiemann series theorem, any conditionally convergent series can be permuted so that its sum is any finite real number or so that it diverges. When an absolutely convergent series is rearranged, its sum is always preserved.

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 $a_{n}$ are not numbers but rather elements of an arbitraryabelian topological group. In that case, instead of using theabsolute value, the definition requires the group to have anorm, which is a positive real-valued function ${\textstyle \|\cdot \|:G\to \mathbb {R} _{+}}$ on an abelian group $G {\displaystyle G}$ (writtenadditively, withidentity element 0) such that:

The norm of the identity element of $G {\displaystyle G}$ is zero: $\|0\|=0.$
For every $x\in G,$ $\|x\|=0$ implies $x=0.$
For every $x\in G,$ $\|-x\|=\|x\|.$
For every $x,y\in G,$ $\|x+y\|\leq \|x\|+\|y\|.$

In this case, the function $d(x,y)=\|x-y\|$ induces the structure of ametric space (a type oftopology) on $G . {\displaystyle G.}$

Then, a $G {\displaystyle G}$ -valued series is absolutely convergent if ${\textstyle \sum _{n=0}^{\infty }\|a_{n}\|<\infty .}$

In particular, these statements apply using the norm $|x|$ (absolute value) in the space of real numbers or complex numbers.

In topological vector spaces

If $X {\displaystyle X}$ is atopological vector space (TVS) and ${\textstyle \left(x_{\alpha }\right)_{\alpha \in A}}$ is a (possiblyuncountable) family in $X {\displaystyle X}$ then this family isabsolutely summable if^[1]

${\textstyle \left(x_{\alpha }\right)_{\alpha \in A}}$ issummable in $X {\displaystyle X}$ (that is, if the limit ${\textstyle \lim _{H\in {\mathcal {F}}(A)}x_{H}}$ of thenet $\left(x_{H}\right)_{H\in {\mathcal {F}}(A)}$ converges in $X, {\displaystyle X,}$ where ${\mathcal {F}}(A)$ is thedirected set of all finite subsets of $A {\displaystyle A}$ directed by inclusion $\subseteq$ and ${\textstyle x_{H}:=\sum _{i\in H}x_{i}}$ ), and
for every continuousseminorm $p {\displaystyle p}$ on $X, {\displaystyle X,}$ the family ${\textstyle \left(p\left(x_{\alpha }\right)\right)_{\alpha \in A}}$ is summable in $\mathbb {R} .$

If $X {\displaystyle X}$ is a normable space and if ${\textstyle \left(x_{\alpha }\right)_{\alpha \in A}}$ is an absolutely summable family in $X, {\displaystyle X,}$ then necessarily all but a countable collection of $x_{\alpha }$ 's are 0.

Absolutely summable families play an important role in the theory ofnuclear spaces.

Relation to convergence

If $G {\displaystyle G}$ iscomplete with respect to the metric $d, {\displaystyle d,}$ 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 anyBanach 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 calledconditionally convergent. An example of a conditionally convergent series is thealternating harmonic series. Many standard tests for divergence and convergence, most notably including theratio test and theroot test, demonstrate absolute convergence. This is because apower series is absolutely convergent on the interior of its disk of convergence.^[a]

Proof that any absolutely convergent series of complex numbers is convergent

Suppose that ${\textstyle \sum \left|a_{k}\right|,a_{k}\in \mathbb {C} }$ is convergent. Then equivalently, ${\textstyle \sum \left[\operatorname {Re} \left(a_{k}\right)^{2}+\operatorname {Im} \left(a_{k}\right)^{2}\right]^{1/2}}$ is convergent, which implies that ${\textstyle \sum \left|\operatorname {Re} \left(a_{k}\right)\right|}$ and ${\textstyle \sum \left|\operatorname {Im} \left(a_{k}\right)\right|}$ converge by termwise comparison of non-negative terms. It suffices to show that the convergence of these series implies the convergence of ${\textstyle \sum \operatorname {Re} \left(a_{k}\right)}$ and ${\textstyle \sum \operatorname {Im} \left(a_{k}\right),}$ for then, the convergence of ${\textstyle \sum a_{k}=\sum \operatorname {Re} \left(a_{k}\right)+i\sum \operatorname {Im} \left(a_{k}\right)}$ would follow, by the definition of the convergence of complex-valued series.

The preceding discussion shows that we need only prove that convergence of ${\textstyle \sum \left|a_{k}\right|,a_{k}\in \mathbb {R} }$ implies the convergence of ${\textstyle \sum a_{k}.}$

Let ${\textstyle \sum \left|a_{k}\right|,a_{k}\in \mathbb {R} }$ be convergent. Since $0\leq a_{k}+\left|a_{k}\right|\leq 2\left|a_{k}\right|,$ we have $0\leq \sum _{k=1}^{n}(a_{k}+\left|a_{k}\right|)\leq \sum _{k=1}^{n}2\left|a_{k}\right|.$ Since ${\textstyle \sum 2\left|a_{k}\right|}$ is convergent, ${\textstyle s_{n}=\sum _{k=1}^{n}\left(a_{k}+\left|a_{k}\right|\right)}$ is abounded monotonic sequence of partial sums, and ${\textstyle \sum \left(a_{k}+\left|a_{k}\right|\right)}$ must also converge. Noting that ${\textstyle \sum a_{k}=\sum \left(a_{k}+\left|a_{k}\right|\right)-\sum \left|a_{k}\right|}$ 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 thetriangle inequality.^[2] By theCauchy criterion, ${\textstyle \sum |a_{i}|}$ converges if and only if for any $\varepsilon >0,$ there exists $N {\displaystyle N}$ such that ${\textstyle \left|\sum _{i=m}^{n}\left|a_{i}\right|\right|=\sum _{i=m}^{n}|a_{i}|<\varepsilon }$ for any $n>m\geq N.$ But the triangle inequality implies that ${\textstyle {\big |}\sum _{i=m}^{n}a_{i}{\big |}\leq \sum _{i=m}^{n}|a_{i}|,}$ so that ${\textstyle \left|\sum _{i=m}^{n}a_{i}\right|<\varepsilon }$ for any $n>m\geq N,$ 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 everyBanach space $(X,\|\,\cdot \,\|).$ Let ${\textstyle \sum x_{n}}$ be an absolutely convergent series in $X . {\displaystyle X.}$ As ${\textstyle \sum _{k=1}^{n}\|x_{k}\|}$ is aCauchy sequence of real numbers, for any $\varepsilon >0$ and large enoughnatural numbers $m>n$ it holds: $\left|\sum _{k=1}^{m}\|x_{k}\|-\sum _{k=1}^{n}\|x_{k}\|\right|=\sum _{k=n+1}^{m}\|x_{k}\|<\varepsilon .$

By the triangle inequality for the normǁ⋅ǁ, one immediately gets: $\left\|\sum _{k=1}^{m}x_{k}-\sum _{k=1}^{n}x_{k}\right\|=\left\|\sum _{k=n+1}^{m}x_{k}\right\|\leq \sum _{k=n+1}^{m}\|x_{k}\|<\varepsilon ,$ which means that ${\textstyle \sum _{k=1}^{n}x_{k}}$ is a Cauchy sequence in $X, {\displaystyle X,}$ hence the series is convergent in $X . {\displaystyle X.}$ ^[3]

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.

TheRiemann 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 termunconditional 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 $G {\displaystyle G}$ , as long as $G {\displaystyle G}$ is complete, every series which converges absolutely also converges unconditionally.

Stated more formally:

Theorem— Let $G {\displaystyle G}$ be a normed abelian group. Suppose $\sum _{i=1}^{\infty }a_{i}=A\in G,\quad \sum _{i=1}^{\infty }\|a_{i}\|<\infty .$ If $\sigma :\mathbb {N} \to \mathbb {N}$ is any permutation, then $\sum _{i=1}^{\infty }a_{\sigma (i)}=A.$

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 $G {\displaystyle G}$ , the converse does not always hold: there can exist series which are not absolutely convergent, yet unconditionally convergent.

For example, in theBanach space ℓ^∞, one series which is unconditionally convergent but not absolutely convergent is: $\sum _{n=1}^{\infty }{\tfrac {1}{n}}e_{n},$

where $\{e_{n}\}_{n=1}^{\infty }$ is an orthonormal basis. A theorem ofA. Dvoretzky andC. A. Rogers asserts that every infinite-dimensional Banach space has an unconditionally convergent series that is not absolutely convergent.^[4]

Proof of the theorem

For any $\varepsilon >0,$ we can choose some $\kappa _{\varepsilon },\lambda _{\varepsilon }\in \mathbb {N} ,$ such that: ${\begin{aligned}{\text{ for all }}N>\kappa _{\varepsilon }&\quad \sum _{n=N}^{\infty }\|a_{n}\|<{\tfrac {\varepsilon }{2}}\\{\text{ for all }}N>\lambda _{\varepsilon }&\quad \left\|\sum _{n=1}^{N}a_{n}-A\right\|<{\tfrac {\varepsilon }{2}}\end{aligned}}$

Let ${\begin{aligned}N_{\varepsilon }&=\max \left\{\kappa _{\varepsilon },\lambda _{\varepsilon }\right\}\\M_{\sigma ,\varepsilon }&=\max \left\{\sigma ^{-1}\left(\left\{1,\ldots ,N_{\varepsilon }\right\}\right)\right\}\end{aligned}}$ where $\sigma ^{-1}\left(\left\{1,\ldots ,N_{\varepsilon }\right\}\right)=\left\{\sigma ^{-1}(1),\ldots ,\sigma ^{-1}\left(N_{\varepsilon }\right)\right\}$ so that $M_{\sigma ,\varepsilon }$ is the smallest natural number such that the list $a_{\sigma (1)},\ldots ,a_{\sigma \left(M_{\sigma ,\varepsilon }\right)}$ includes all of the terms $a_{1},\ldots ,a_{N_{\varepsilon }}$ (and possibly others).

Finally for anyinteger $N>M_{\sigma ,\varepsilon }$ let ${\begin{aligned}I_{\sigma ,\varepsilon }&=\left\{1,\ldots ,N\right\}\setminus \sigma ^{-1}\left(\left\{1,\ldots ,N_{\varepsilon }\right\}\right)\\S_{\sigma ,\varepsilon }&=\min \sigma \left(I_{\sigma ,\varepsilon }\right)=\min \left\{\sigma (k)\ :\ k\in I_{\sigma ,\varepsilon }\right\}\\L_{\sigma ,\varepsilon }&=\max \sigma \left(I_{\sigma ,\varepsilon }\right)=\max \left\{\sigma (k)\ :\ k\in I_{\sigma ,\varepsilon }\right\}\\\end{aligned}}$ so that ${\begin{aligned}\left\|\sum _{i\in I_{\sigma ,\varepsilon }}a_{\sigma (i)}\right\|&\leq \sum _{i\in I_{\sigma ,\varepsilon }}\left\|a_{\sigma (i)}\right\|\\&\leq \sum _{j=S_{\sigma ,\varepsilon }}^{L_{\sigma ,\varepsilon }}\left\|a_{j}\right\|&&{\text{ since }}\sigma (I_{\sigma ,\varepsilon })\subseteq \left\{S_{\sigma ,\varepsilon },S_{\sigma ,\varepsilon }+1,\ldots ,L_{\sigma ,\varepsilon }\right\}\\&\leq \sum _{j=N_{\varepsilon }+1}^{\infty }\left\|a_{j}\right\|&&{\text{ since }}S_{\sigma ,\varepsilon }\geq N_{\varepsilon }+1\\&<{\frac {\varepsilon }{2}}\end{aligned}}$ and thus ${\begin{aligned}\left\|\sum _{i=1}^{N}a_{\sigma (i)}-A\right\|&=\left\|\sum _{i\in \sigma ^{-1}\left(\{1,\dots ,N_{\varepsilon }\}\right)}a_{\sigma (i)}-A+\sum _{i\in I_{\sigma ,\varepsilon }}a_{\sigma (i)}\right\|\\&\leq \left\|\sum _{j=1}^{N_{\varepsilon }}a_{j}-A\right\|+\left\|\sum _{i\in I_{\sigma ,\varepsilon }}a_{\sigma (i)}\right\|\\&<\left\|\sum _{j=1}^{N_{\varepsilon }}a_{j}-A\right\|+{\frac {\varepsilon }{2}}\\&<\varepsilon \end{aligned}}$

This shows that ${\text{ for all }}\varepsilon >0,{\text{ there exists }}M_{\sigma ,\varepsilon },{\text{ for all }}N>M_{\sigma ,\varepsilon }\quad \left\|\sum _{i=1}^{N}a_{\sigma (i)}-A\right\|<\varepsilon ,$ that is: $\sum _{i=1}^{\infty }a_{\sigma (i)}=A.$

Products of series

TheCauchy product of two series converges to the product of the sums if at least one of the series converges absolutely. That is, suppose that $\sum _{n=0}^{\infty }a_{n}=A\quad {\text{ and }}\quad \sum _{n=0}^{\infty }b_{n}=B.$

The Cauchy product is defined as the sum of terms $c_{n}$ where: $c_{n}=\sum _{k=0}^{n}a_{k}b_{n-k}.$

Ifeither the $a_{n}$ or $b_{n}$ sum converges absolutely then $\sum _{n=0}^{\infty }c_{n}=AB.$

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 $X {\displaystyle X}$ and a function $f:X\to \mathbb {R} .$ We will give a definition below of the sum of $f {\displaystyle f}$ over $X, {\displaystyle X,}$ written as ${\textstyle \sum _{x\in X}f(x).}$

First note that because no particular enumeration (or "indexing") of $X {\displaystyle X}$ has yet been specified, the series ${\textstyle \sum _{x\in X}f(x)}$ cannot be understood by the more basic definition of a series. In fact, for certain examples of $X {\displaystyle X}$ and $f, {\displaystyle f,}$ the sum of $f {\displaystyle f}$ over $X {\displaystyle X}$ may not be defined at all, since some indexing may produce a conditionally convergent series.

Therefore we define ${\textstyle \sum _{x\in X}f(x)}$ only in the case where there exists some bijection $g:\mathbb {Z} ^{+}\to X$ such that ${\textstyle \sum _{n=1}^{\infty }f(g(n))}$ is absolutely convergent. Note that here, "absolutely convergent" uses the more basic definition, applied to an indexed series. In this case, the value of thesum of $f {\displaystyle f}$ over $X {\displaystyle X}$ ^[5] is defined by $\sum _{x\in X}f(x):=\sum _{n=1}^{\infty }f(g(n))$

Note that because the series is absolutely convergent, then every rearrangement is identical to a different choice of bijection $g . {\displaystyle g.}$ Since all of these sums have the same value, then the sum of $f {\displaystyle f}$ over $X {\displaystyle X}$ is well-defined.

Even more generally we may define the sum of $f {\displaystyle f}$ over $X {\displaystyle X}$ when $X {\displaystyle X}$ is uncountable. But first we define what it means for the sum to be convergent.

Let $X {\displaystyle X}$ be any set, countable or uncountable, and $f:X\to \mathbb {R}$ a function. We say thatthe sum of $f {\displaystyle f}$ over $X {\displaystyle X}$ converges absolutely if $\sup \left\{\sum _{x\in A}|f(x)|:A\subseteq X,A{\text{ is finite }}\right\}<\infty .$

There is a theorem which states that, if the sum of $f {\displaystyle f}$ over $X {\displaystyle X}$ is absolutely convergent, then $f {\displaystyle f}$ takes non-zero values on a set that is at most countable. Therefore, the following is a consistent definition of the sum of $f {\displaystyle f}$ over $X {\displaystyle X}$ when the sum is absolutely convergent. $\sum _{x\in X}f(x):=\sum _{x\in X:f(x)\neq 0}f(x).$

Note that the final series uses the definition of a series over acountable set.

Some authors define an iterated sum ${\textstyle \sum _{m=1}^{\infty }\sum _{n=1}^{\infty }a_{m,n}}$ to be absolutely convergent if the iterated series ${\textstyle \sum _{m=1}^{\infty }\sum _{n=1}^{\infty }|a_{m,n}|<\infty .}$ ^[6] This is in fact equivalent to the absolute convergence of ${\textstyle \sum _{(m,n)\in \mathbb {N} \times \mathbb {N} }a_{m,n}.}$ That is to say, if the sum of $f {\displaystyle f}$ over $X, {\displaystyle X,}$ ${\textstyle \sum _{(m,n)\in \mathbb {N} \times \mathbb {N} }a_{m,n},}$ converges absolutely, as defined above, then the iterated sum ${\textstyle \sum _{m=1}^{\infty }\sum _{n=1}^{\infty }a_{m,n}}$ converges absolutely, and vice versa.

Absolute convergence of integrals

Theintegral ${\textstyle \int _{A}f(x)\,dx}$ of a real or complex-valued function is said toconverge absolutely if ${\textstyle \int _{A}\left|f(x)\right|\,dx<\infty .}$ One also says that $f {\displaystyle f}$ isabsolutely integrable. The issue of absolute integrability is intricate and depends on whether theRiemann,Lebesgue, orKurzweil-Henstock (gauge) integral is considered; for the Riemann integral, it also depends on whether we only consider integrability in its proper sense ( $f {\displaystyle f}$ and $A {\displaystyle A}$ bothbounded), or permit the more general case of improper integrals.

As a standard property of the Riemann integral, when $A=[a,b]$ is a boundedinterval, everycontinuous function is bounded and (Riemann) integrable, and since $f {\displaystyle f}$ continuous implies $|f|$ continuous, every continuous function is absolutely integrable. In fact, since $g\circ f$ is Riemann integrable on $[a,b]$ if $f {\displaystyle f}$ is (properly) integrable and $g {\displaystyle g}$ is continuous, it follows that $|f|=|\cdot |\circ f$ is properly Riemann integrable if $f {\displaystyle f}$ is. However, this implication does not hold in the case of improper integrals. For instance, the function ${\textstyle f:[1,\infty )\to \mathbb {R} :x\mapsto {\frac {\sin x}{x}}}$ is improperly Riemann integrable on its unbounded domain, but it is not absolutely integrable: $\int _{1}^{\infty }{\frac {\sin x}{x}}\,dx={\frac {1}{2}}{\bigl [}\pi -2\,\mathrm {Si} (1){\bigr ]}\approx 0.62,{\text{ but }}\int _{1}^{\infty }\left|{\frac {\sin x}{x}}\right|dx=\infty .$ Indeed, more generally, given any series ${\textstyle \sum _{n=0}^{\infty }a_{n}}$ one can consider the associatedstep function $f_{a}:[0,\infty )\to \mathbb {R}$ defined by $f_{a}([n,n+1))=a_{n}.$ Then ${\textstyle \int _{0}^{\infty }f_{a}\,dx}$ converges absolutely, converges conditionally or diverges according to the corresponding behavior of ${\textstyle \sum _{n=0}^{\infty }a_{n}.}$

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 $|f|$ is unbounded in the examples above implies that $f {\displaystyle f}$ is also not integrable in the Lebesgue sense. In fact, in the Lebesgue theory of integration, given that $f {\displaystyle f}$ ismeasurable, $f {\displaystyle f}$ is (Lebesgue) integrable if and only if $|f|$ is (Lebesgue) integrable. However, the hypothesis that $f {\displaystyle f}$ is measurable is crucial; it is not generally true that absolutely integrable functions on $[a,b]$ are integrable (simply because they may fail to be measurable): let $S\subset [a,b]$ be a nonmeasurablesubset and consider $f=\chi _{S}-1/2,$ where $\chi _{S}$ is thecharacteristic function of $S . {\displaystyle S.}$ Then $f {\displaystyle f}$ is not Lebesgue measurable and thus not integrable, but $|f|\equiv 1/2$ is a constant function and clearly integrable.

On the other hand, a function $f {\displaystyle f}$ may be Kurzweil-Henstock integrable (gauge integrable) while $|f|$ is not. This includes the case of improperly Riemann integrable functions.

In a general sense, on anymeasure space $A, {\displaystyle A,}$ the Lebesgue integral of a real-valued function is defined in terms of its positive and negative parts, so the facts:

$f {\displaystyle f}$ integrable implies $|f|$ integrable
$f {\displaystyle f}$ measurable, $|f|$ integrable implies $f {\displaystyle f}$ integrable

are essentially built into the definition of the Lebesgue integral. In particular, applying the theory to thecounting measure on aset $S, {\displaystyle S,}$ one recovers the notion of unordered summation of series developed by Moore–Smith using (what are now called) nets. When $S=\mathbb {N}$ 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 morefunctional analytic approach, obtaining theBochner integral.

See also

Cauchy principal value – Method for assigning values to certain improper integrals which would otherwise be undefined
Conditional convergence – Property of infinite series
Convergence of Fourier series – Mathematical problem in classical harmonic analysis
Fubini's theorem – Conditions for switching order of integration in calculus
Modes of convergence (annotated index) – Annotated index of various modes of convergence
Radius of convergence – Domain of convergence of power series
Riemann series theorem – Unconditionally convergent series converge absolutely
Unconditional convergence – Order-independent convergence of a sequence
1/2 − 1/4 + 1/8 − 1/16 + · · · – Infinite series summable to 1/3Pages displaying short descriptions of redirect targets
1/2 + 1/4 + 1/8 + 1/16 + · · · – Infinite series summable to 1Pages displaying short descriptions of redirect targets

Notes

^Here, the disk of convergence is used to refer to all points whose distance from the center of the series is less than the radius of convergence. That is, the disk of convergence is made up of all points for which the power series converges.

References

^Schaefer, Helmut H.; Wolff, Manfred P. (1999).Topological Vector Spaces.GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. pp. 179–180.ISBN 978-1-4612-7155-0.OCLC 840278135.
^Rudin, Walter (1976).Principles of Mathematical Analysis. New York: McGraw-Hill. pp. 71–72.ISBN 0-07-054235-X.
^Megginson, Robert E. (1998),An introduction to Banach space theory, Graduate Texts in Mathematics, vol. 183, New York: Springer-Verlag, p. 20,ISBN 0-387-98431-3 (Theorem 1.3.9)
^Dvoretzky, A.; Rogers, C. A. (1950), "Absolute and unconditional convergence in normed linear spaces", Proc. Natl. Acad. Sci. U.S.A.36:192–197.
^Tao, Terence (2016).Analysis I. New Delhi: Hindustan Book Agency. pp. 188–191.ISBN 978-9380250649.
^Strichartz, Robert (2000).The Way of Analysis. Jones & Bartlett Learning. pp. 259–260.ISBN 978-0763714970.

General references

Narici, Lawrence; Beckenstein, Edward (2011).Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.ISBN 978-1584888666.OCLC 144216834.
Walter Rudin,Principles of Mathematical Analysis (McGraw-Hill: New York, 1964).
Pietsch, Albrecht (1979).Nuclear Locally Convex Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 66 (Second ed.). Berlin, New York:Springer-Verlag.ISBN 978-0-387-05644-9.OCLC 539541.
Robertson, A. P. (1973).Topological vector spaces. Cambridge England: University Press.ISBN 0-521-29882-2.OCLC 589250.
Ryan, Raymond A. (2002).Introduction to Tensor Products of Banach Spaces.Springer Monographs in Mathematics. London New York:Springer.ISBN 978-1-85233-437-6.OCLC 48092184.
Trèves, François (2006) [1967].Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications.ISBN 978-0-486-45352-1.OCLC 853623322.
Wong, Yau-Chuen (1979).Schwartz Spaces, Nuclear Spaces, and Tensor Products.Lecture Notes in Mathematics. Vol. 726. Berlin New York:Springer-Verlag.ISBN 978-3-540-09513-2.OCLC 5126158.

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Sequences andseries

List of mathematical series

Integer sequences

Basic	Arithmetic progression Geometric progression Harmonic progression Square number Cubic number Factorial Powers of two Powers of three Powers of 10
Advanced(list)	Complete sequence Fibonacci sequence Figurate number Heptagonal number Hexagonal number Lucas number Pell number Pentagonal number Polygonal number Triangular number array

Fibonacci spiral with square sizes up to 34.

Properties of sequences

Properties of series

Series	Alternating Convergent Divergent Telescoping
Convergence	Absolute Conditional Uniform

Explicit series

Convergent	1/2 − 1/4 + 1/8 − 1/16 + ⋯ 1/2 + 1/4 + 1/8 + 1/16 + ⋯ 1/4 + 1/16 + 1/64 + 1/256 + ⋯ 1 + 1/2^s + 1/3^s + ... (Riemann zeta function)
Divergent	1 + 1 + 1 + 1 + ⋯ 1 − 1 + 1 − 1 + ⋯ (Grandi's series) 1 + 2 + 3 + 4 + ⋯ 1 − 2 + 3 − 4 + ⋯ 1 + 2 + 4 + 8 + ⋯ 1 − 2 + 4 − 8 + ⋯ Infinite arithmetic series 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials) 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series) 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)

Kinds of series

Hypergeometric series

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