Conditional convergence
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 |
Conditional convergence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conditional_convergence&oldid=19647