Absolute convergence


In mathematics, an infinite series of numbers is said to converge absolutely 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 A convergent series that is not absolutely convergent is called conditionally 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 both associative and commutative, meaning that grouping and rearrangement do not alter the final sum. For instance, is equal to both and. However, associativity and commutativity do not necessarily hold for infinite sums. One example is the alternating harmonic series
whose terms are fractions that alternate in sign. This series is convergent and can be evaluated using the Maclaurin series for the function, which converges for all satisfying :
Substituting reveals that the original sum is equal to. The sum can also be rearranged as follows:
In this rearrangement, the reciprocal of each odd 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
or half the original series. The violation of the associativity and commutativity of addition reveals that the alternating harmonic series is conditionally convergent. Indeed, the sum of the absolute values of each term is, or the divergent harmonic series. According to the Riemann 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 is absolutely convergent if the sum of the absolute values of the terms converges.

Sums of more general elements

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 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 on
Then, a -valued series is absolutely convergent if
In particular, these statements apply using the norm in the space of real numbers or complex numbers.

In topological vector spaces

If is a topological vector space and is a family in then this family is absolutely summable if
  1. is summable in , 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 convergence

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.

Proof that any absolutely convergent series of complex numbers is convergent

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 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, 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 convergent

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, one immediately gets:
which means that is a Cauchy sequence in hence the series is convergent in

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, as long as 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, 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:
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.

Proof of the theorem

For any we can choose some such that:
Let
where so that is the smallest natural number such that the list includes all of the terms .
Finally for any integer let
so that
and thus
This shows that
that is:
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
The Cauchy product is defined as the sum of terms where:
If the or sum converges absolutely then

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 and a function We will give a definition below of the sum of over written as
First note that because no particular enumeration 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 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 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.