Absolute convergence

From Wikipedia, the free encyclopedia

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

$\sum_{n=1}^\infty a_n$

is said to converge absolutely if

$\sum_{n=1}^\infty \left|a_n\right|<\infty.$

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 the integral of the corresponding absolute value is finite, i.e.

$\int_A \left|f(x)\right|\,dx<\infty.$

1 Rearrangements
2 Products of series
3 Conditional convergence
4 See also
5 References

[edit] 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

$\sum_{n=1}^\infty a_{\sigma(n)}$

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.

[edit] 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

$\sum_{n=0}^\infty c_n = AB$

[edit] Conditional convergence

A conditionally convergent series or integral is one that converges but does not converge absolutely. An example is

$1 - {1 \over 2} + {1 \over 3} - {1 \over 4} + {1 \over 5} - \cdots =\sum_{n=1}^\infty (-1)^{n+1} {1 \over n}$

which converges to log_e 2, but is not absolutely convergent (see Harmonic series).

Bernhard Riemann proved that a conditionally convergent series may be rearranged to converge to any sum at all, including ∞ or −∞; see Riemann series theorem.

[edit] See also

[edit] References

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

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Absolute convergence

From Wikipedia, the free encyclopedia

Contents

[edit] Rearrangements

[edit] Products of series

[edit] Conditional convergence

[edit] See also

[edit] References

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