Absolute convergence

Summary

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 is said to converge absolutely if for some real number Similarly, an improper integral of a function, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if

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 converges to while its rearrangement (in which the repeating pattern of signs is two positive terms followed by one negative term) converges to

BackgroundEdit

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

 

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:

 

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:

 

This leads to an apparent paradox: does   or  ?

The answer is that because S is not absolutely convergent, rearranging its terms changes the value of the sum. This means   and   are not equal. In fact, the series   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.

ExplanationEdit

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

 

or with equal possibility, that

 

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:

 
 

and in the second case:

 
 

This reveals the trick: the definition of S was interpreted as defining its last term as negative when evaluating   but positive when evaluating   when in fact the definition of S didn't define (and the rearrangement was independent of) either option.

Definition for real and complex numbersEdit

A sum of real numbers or complex numbers   is absolutely convergent if the sum of the absolute values of the terms   converges.

Sums of more general elementsEdit

The same definition can be used for series   whose terms   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   on an abelian group   (written additively, with identity element 0) such that:

  1. The norm of the identity element of   is zero:  
  2. For every     implies  
  3. For every    
  4. For every    

In this case, the function   induces the structure of a metric space (a type of topology) on  

Then, a  -valued series is absolutely convergent if  

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

In topological vector spacesEdit

If   is a topological vector space (TVS) and   is a (possibly uncountable) family in   then this family is absolutely summable if[1]

  1.   is summable in   (that is, if the limit   of the net   converges in   where   is the directed set of all finite subsets of   directed by inclusion   and  ), and
  2. for every continuous seminorm   on   the family   is summable in  

If   is a normable space and if   is an absolutely summable family in   then necessarily all but a countable collection of  's are 0.

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

Relation to convergenceEdit

If   is complete with respect to the metric   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.[a]

Proof that any absolutely convergent series of complex numbers is convergentEdit

Suppose that   is convergent. Then equivalently,   is convergent, which implies that   and   converge by termwise comparison of non-negative terms. It suffices to show that the convergence of these series implies the convergence of   and   for then, the convergence of   would follow, by the definition of the convergence of complex-valued series.

The preceding discussion shows that we need only prove that convergence of   implies the convergence of  

Let   be convergent. Since   we have

 
Since   is convergent,   is a bounded monotonic sequence of partial sums, and   must also converge. Noting that   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 inequalityEdit

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.[2] By the Cauchy criterion,   converges if and only if for any   there exists   such that   for any   But the triangle inequality implies that   so that   for any   which is exactly the Cauchy criterion for  

Proof that any absolutely convergent series in a Banach space is convergentEdit

The above result can be easily generalized to every Banach space   Let   be an absolutely convergent series in   As   is a Cauchy sequence of real numbers, for any   and large enough natural numbers   it holds:

 

By the triangle inequality for the norm ǁ⋅ǁ, one immediately gets:

 
which means that   is a Cauchy sequence in   hence the series is convergent in  [3]

Rearrangements and unconditional convergenceEdit

Real and complex numbersEdit

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 spaceEdit

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  , as long as   is complete, every series which converges absolutely also converges unconditionally.

Stated more formally:

Theorem —  Let   be a normed abelian group. Suppose

 
If   is any permutation, then
 

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  , the converse does not always hold: there can exist series which are not absolutely convergent, yet unconditionally convergent.

For example, in the Banach space, one series which is unconditionally convergent but not absolutely convergent is:

 

where   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.[4]

Proof of the theoremEdit

For any   we can choose some   such that:

 

Let

 

Finally for any integer   let

 

Then

 

This shows that

 
that is:
 

Q.E.D.

Products of seriesEdit

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

 

The Cauchy product is defined as the sum of terms   where:

 

If either the   or   sum converges absolutely then

 

Absolute convergence over setsEdit

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   and a function   We will give a definition below of the sum of   over   written as  

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

Therefore we define   only in the case where there exists some bijection   such that   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   over  [5] is defined by

 

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

Even more generally we may define the sum of   over   when   is uncountable. But first we define what it means for the sum to be convergent.

Let   be any set, countable or uncountable, and   a function. We say that the sum of   over   converges absolutely if

 

There is a theorem which states that, if the sum of   over   is absolutely convergent, then   takes non-zero values on a set that is at most countable. Therefore, the following is a consistent definition of the sum of   over   when the sum is absolutely convergent.

 

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

Some authors define an iterated sum   to be absolutely convergent if the iterated series  [6] This is in fact equivalent to the absolute convergence of   That is to say, if the sum of   over     converges absolutely, as defined above, then the iterated sum   converges absolutely, and vice versa.

Absolute convergence of integralsEdit

The integral   of a real or complex-valued function is said to converge absolutely if   One also says that   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 (  and   both bounded), or permit the more general case of improper integrals.

As a standard property of the Riemann integral, when   is a bounded interval, every continuous function is bounded and (Riemann) integrable, and since   continuous implies   continuous, every continuous function is absolutely integrable. In fact, since   is Riemann integrable on   if   is (properly) integrable and   is continuous, it follows that   is properly Riemann integrable if   is. However, this implication does not hold in the case of improper integrals. For instance, the function   is improperly Riemann integrable on its unbounded domain, but it is not absolutely integrable:

 
Indeed, more generally, given any series   one can consider the associated step function   defined by   Then   converges absolutely, converges conditionally or diverges according to the corresponding behavior of  

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   is unbounded in the examples above implies that   is also not integrable in the Lebesgue sense. In fact, in the Lebesgue theory of integration, given that   is measurable,   is (Lebesgue) integrable if and only if   is (Lebesgue) integrable. However, the hypothesis that   is measurable is crucial; it is not generally true that absolutely integrable functions on   are integrable (simply because they may fail to be measurable): let   be a nonmeasurable subset and consider   where   is the characteristic function of   Then   is not Lebesgue measurable and thus not integrable, but   is a constant function and clearly integrable.

On the other hand, a function   may be Kurzweil-Henstock integrable (gauge integrable) while   is not. This includes the case of improperly Riemann integrable functions.

In a general sense, on any measure space   the Lebesgue integral of a real-valued function is defined in terms of its positive and negative parts, so the facts:

  1.   integrable implies   integrable
  2.   measurable,   integrable implies   integrable

are essentially built into the definition of the Lebesgue integral. In particular, applying the theory to the counting measure on a set   one recovers the notion of unordered summation of series developed by Moore–Smith using (what are now called) nets. When   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.

See alsoEdit

NotesEdit

  1. ^ 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.

ReferencesEdit

  1. ^ Schaefer & Wolff 1999, pp. 179–180.
  2. ^ Rudin, Walter (1976). Principles of Mathematical Analysis. New York: McGraw-Hill. pp. 71–72. ISBN 0-07-054235-X.
  3. ^ 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)
  4. ^ Dvoretzky, A.; Rogers, C. A. (1950), "Absolute and unconditional convergence in normed linear spaces", Proc. Natl. Acad. Sci. U.S.A. 36:192–197.
  5. ^ Tao, Terrance (2016). Analysis I. New Delhi: Hindustan Book Agency. pp. 188–191. ISBN 978-9380250649.
  6. ^ Strichartz, Robert (2000). The Way of Analysis. Jones & Bartlett Learning. pp. 259, 260. ISBN 978-0763714970.

Works citedEdit

  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.

General referencesEdit

  • 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.