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Conditional convergence

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of a series

A property of series, stating that the given series converges after a certain (possibly trivial) rearrangement of its terms. A series of numbers

(*)

is unconditionally convergent if it converges itself, as well as any series obtained by rearranging its terms, while the sum of any such series is the same; in other words: The sum of an unconditionally-convergent series does not depend on the order of its terms. If the series (*) converges, but not unconditionally, then it is said to be conditionally convergent. For the series (*) to be conditionally convergent it is necessary and sufficient that it converges and does not absolutely converge, i.e. that .

If the terms of the series (*) are real numbers, if the non-negative terms are denoted by and the negative terms by then the series (*) is conditionally convergent if and only both series and diverge (here the order of the terms in the series is immaterial).

Let the series (*) of real numbers be conditionally convergent and let , then there exists a series , obtained by rearranging the terms of (*), such that if denotes its sequence of partial sums, then

(this is a generalization of Riemann's theorem, cf. Riemann theorem 2).

The product of two conditionally-convergent series depends on the order in which the result of the term-by-term multiplication of the two series is summed.

The concepts of conditional and unconditional convergence of series may be generalized to series with terms in some normed vector space . If is a finite-dimensional space then, analogously to the case of series of numbers, a convergent series , , is conditionally convergent if and only if the series is divergent. If, however, is infinite dimensional, then there exist unconditionally-convergent series .


Comments

A very useful reference on convergence and divergence of series with elements in abstract spaces is [a1].

References

[a1] J. Lindenstrauss, L. Tzafriri, "Classical Banach spaces" , 1. Sequence spaces , Springer (1977)
[a2] W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 107–108
How to Cite This Entry:
Conditional convergence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conditional_convergence&oldid=19008
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article