For instance, rearrangements do not change the value of the sum, which is not necessarily true for conditionally convergent series.
When adding a finite number of terms, addition is both associative and commutative, meaning that grouping and rearrangment do not alter the final sum.
However, associativity and commutativity do not necessarily hold for infinite sums.
The violation of the associativity and commutativity of addition reveals that the alternating harmonic series is conditionally convergent.
When an absolutely convergent series is rearranged, its sum is always preserved.
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
(written additively, with identity element 0) such that: In this case, the function
induces the structure of a metric space (a type of topology) on
Absolutely summable families play an important role in the theory of nuclear spaces.
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.
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.
[3] 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.
For any series with values in a normed abelian group
However, in the more general case of a series with values in any normed abelian group
may not be defined at all, since some indexing may produce a conditionally convergent series.
Note that here, "absolutely convergent" uses the more basic definition, applied to an indexed series.
Note that because the series is absolutely convergent, then every rearrangement is identical to a different choice of bijection
both bounded), or permit the more general case of improper integrals.
However, this implication does not hold in the case of improper integrals.
is measurable is crucial; it is not generally true that absolutely integrable functions on
This includes the case of improperly Riemann integrable functions.
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.
one recovers the notion of unordered summation of series developed by Moore–Smith using (what are now called) nets.
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.