View source for 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 
$$\sum_{n=1}^\infty u_n\label{*}$$
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 $\sum_{n=1}^\infty |u_n| = +\infty$.

If the terms of the series (*) are real numbers, if the non-negative
terms are denoted by $u_1^+, u_2^+,\dots $ and the negative terms by $-u_1^-, -u_2^-,\dots $ then the series
(*) is conditionally convergent if and only both series $\sum_{n=1}^\infty u_n^+ $ and $\sum_{n=1}^\infty u_n^- $
diverge (here the order of the terms in the series is immaterial).

Let the series (*) of real numbers be conditionally convergent and let
$-\infty\le \alpha < \beta \le +\infty$, then there exists a series $\sum_{n=1}^\infty u_n^*$, obtained by rearranging the terms
of (*), such that if $\{s_n^*\}$ denotes its sequence of partial sums, then

$$\underline{\lim}_{n\to\infty}\; s_n^* = \alpha,\quad \overline{\lim}_{n\to\infty}\; s_n^* = \beta$$
(this is a generalization of Riemann's theorem, cf.
[[Riemann theorem|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
$X$. If $X$ is a finite-dimensional space then, analogously to the
case of series of numbers, a convergent series $\sum_{n=1}^\infty u_n$, $u_n\in X$, $n=1,2,\dots$ is
conditionally convergent if and only if the series $\sum_{n=1}^\infty \|u_n\|_X$ is
divergent. If, however, $X$ is infinite dimensional, then there exist
unconditionally-convergent series $\sum_{n=1}^\infty \|u_n\|_X = +\infty$.



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

====References====
<table><TR><TD valign="top">[a1]</TD>
 <TD valign="top"> J. Lindenstrauss, L. Tzafriri, "Classical Banach spaces" , '''1. Sequence spaces''' , Springer (1977)</TD>
</TR><TR><TD valign="top">[a2]</TD>
 <TD valign="top"> W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 107–108</TD>
</TR></table>