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 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 [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