Absolute convergence

In mathematics, a series (or sometimes also an integral) of numbers is said to converge absolutely if the sum (or integral) of the absolute value of the summand or integrand is finite. More precisely, a real or complex series is said to converge absolutely if
Absolute convergence is important to the study of infinite series because its definition is strong enough to have properties of finite sums that not all convergent series possess, yet is broad enough to occur commonly.
Contents
Background
One may study the convergence of series whose terms a_{n} are elements of an arbitrary abelian topological group. The notion of absolute convergence requires more structure, namely a norm:
A norm on an abelian group G (written additively, with identity element 0) is a realvalued function on G such that:
 The norm of the identity element of G is zero:
 For every x in G,
 For every x in G,
 For every x, y in G,
Then the function induces on G the structure of a metric space (in particular, a topology). We can therefore consider Gvalued series and define such a series to be absolutely convergent if
Relation to convergence
If the metric d on G is complete, then every absolutely convergent series is convergent. The proof is the same as for complexvalued 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 is convergent
Assume is convergent. Since a series of complex numbers converges if and only if both its real and imaginary parts converge, we may assume with equal generality that . Then, is convergent.
Since , we have:
 .
Thus, is a bounded monotonic sequence (in m), which must converge.
is a difference of convergent series; therefore, it is also convergent. is convergent is convergent.
Rearrangements and unconditional convergence
In the general context of a Gvalued series, a distinction is made between absolute and unconditional convergence, and the assertion that a real or complex series which is not absolutely convergent is necessarily conditionally convergent (meaning not unconditionally convergent) is then a theorem, not a definition. This is discussed in more detail below.
Given a series with values in a normed abelian group G and a permutation σ of the natural numbers, one builds a new series , said to be a rearrangement of the original series. A series is said to be unconditionally convergent if all rearrangements of the series are convergent to the same value.
When G is complete, absolute convergence implies unconditional convergence.
Theorem
Let , and let σ be a permutation of , then
Proof
For anyε > 0, we can choose some , such that
and
 .
let N_{ε}: = max(κ_{ε},λ_{ε}), .
For any , let
 ,
 (note.), and
then
therefore
then
The issue of the converse is much more interesting. For real series it follows from the Riemann rearrangement theorem that unconditional convergence implies absolute convergence. Since a series with values in a finitedimensional normed space is absolutely convergent if each of its onedimensional projections is absolutely convergent, it follows easily that absolute and unconditional convergence coincide for valued series.
But there is an unconditionally and nonabsolutely convergent series with values in Hilbert space : if is an orthonormal basis, take .
A theorem of DvoretzkyRogers asserts that every infinitedimensional Banach space admits an unconditionally but nonabsolutely convergent series.
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:
The Cauchy product is defined as the sum of terms c_{n} where:
Then, if either the a_{n} or b_{n} sum converges absolutely, then
Absolute convergence of integrals
The integral of a real or complexvalued function is said to converge absolutely if One also says that f is absolutely integrable.
When A = [a,b] is a closed bounded interval, every continuous function is integrable, and since f continuous implies  f  continuous, similarly every continuous function is absolutely integrable. It is not generally true that absolutely integrable functions on [a,b] are integrable: let be a nonmeasurable subset and take , where χ_{S} is the characteristic function of S. Then f is not Lebesgue measurable but  f  is constant. However, it is a standard result that if f is Riemann integrable, so is  f  . This holds also for the Lebesgue integral; see below. On the other hand a function f may be KurzweilHenstock integrable (or "gauge integrable") while  f  is not. This includes the case of improperly Riemann integrable functions.
Similarly, when A is an interval of infinite length it is wellknown that there are improperly Riemann integrable functions f which are not absolutely integrable. Indeed, given any series one can consider the associated step function defined by f_{a}([n,n + 1)) = a_{n}. Then converges absolutely, converges conditionally or diverges according to the corresponding behavior of
Another example of a convergent but not absolutely convergent improper Riemann integral is .
On any measure space A the Lebesgue integral of a realvalued function is defined in terms of its positive and negative parts, so the facts:
 f integrable implies  f  integrable
 f measurable,  f  integrable implies f integrable
are essentially built into the definition of the Lebesgue integral. In particular, applying the theory to the counting measure on a set S, one recovers the notion of unordered summation of series developed by MooreSmith 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 Banachvalued 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 also
 Convergence of Fourier series
 Conditional convergence
 Modes of convergence (annotated index)
 Cauchy principal value
 A counterexample related to Fubini's theorem
 1/2 − 1/4 + 1/8 − 1/16 + · · ·
 1/2 + 1/4 + 1/8 + 1/16 + · · ·
References
 Walter Rudin, Principles of Mathematical Analysis (McGrawHill: New York, 1964).
Categories: Mathematical series
 Integral calculus
 Summability theory
 Convergence (mathematics)
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