Discover

In mathematics, a series or integral is said to 'converge absolutely' if the sum or integral of the absolute value of the summand or integrand is finite. The property of absolute convergence is important because it is generally required in order for rearrangements and products of sums to work in an intuitive fashion.
More precisely, a series $sum\_\{n=0\}^infty\; a\_n$ is said to 'converge absolutely' if and only if $sum\_\{n=0\}^infty\; left|a\_n$
ight| < infty.
If $a\_n$ is a complex number, this theorem can be imagined as follows: the sum of all $a\_k$ is a vector addition path through the complex plane. If the length of the path, that is the sum of all the lengths of the parts $|a\_k|$, is finite, the end point has to be a finite distance from the origin.
Likewise, an integral $int\_A\; f(x),dx$ is said to 'converge absolutely' if and only if $int\_A\; left|f(x)$
ight|,dx < infty.

Contents

Rearrangements

Products of series

See also

References

Rearrangements

Absolute convergence means that the value of the sum/integral is independent of the order in which the sum is performed. That is, a rearrangement of the series
:$sum\_\{n=0\}^infty\; a\_\{sigma(n)\}$
where σ is a permutation of the natural numbers, does not alter the sum to which the series converges. Similar results apply to integrals.
In the light of Lebesgue's theory of integration, sums may be treated as special cases of integrals, rather than as a separate case.

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:
:$sum\_\{n=0\}^infty\; a\_n\; =\; A$
:$sum\_\{n=0\}^infty\; b\_n\; =\; B.$
The Cauchy product is defined as the sum of terms $c\_n$ where:
:$c\_n\; =\; sum\_\{k=0\}^n\; a\_k\; b\_\{n-k\}.$
Then, if ''either'' the $a\_n$ or $b\_n$ sum converges absolutely, then
:$sum\_\{n=0\}^infty\; c\_n\; =\; AB.$

See also

★ Conditional convergence

★ Cauchy principal value

★ A counterexample related to Fubini's theorem

References

★ Walter Rudin, ''Principles of Mathematical Analysis'' (McGraw-Hill: New York, 1964).