Alternatively, for every chain of finite subsets of , the partial sums
converges. The trick to see this equivalence is to realize two facts: 1. every subsequence of a convergent sequence is convergent, and 2. every chain can be enlarged to a maximal chain , such that . Then the series indexed by is a subseries indexed by , which is a subseries of a permutation of the original convergent series.
Every absolutely convergent series is unconditionally convergent, the converse implication does not hold in general.
When then by a famous theorem of Riemann is unconditionally convergent if and only if it is absolutely convergent.
- 1 K. Knopp: Theory and application of infinite series.
- 2 K. Knopp: Infinite sequences and series.
- 3 P. Wojtaszczyk: Banach spaces for analysts.
- 4 Ch. Heil: http://www.math.gatech.edu/ heil/papers/bases.pdfA basis theory primer.
|Date of creation||2013-03-22 15:29:57|
|Last modified on||2013-03-22 15:29:57|
|Last modified by||kompik (10588)|