A series is said to be convergent if the sequence of partial sums is convergent. A series that is not convergent is said to be divergent.
A series is said to be absolutely convergent if is convergent.
When the terms of the series live in , an equivalent condition for absolute convergence of the series is that all possible series obtained by rearrangements of the terms are also convergent. (This is not true in arbitrary metric spaces.)
Let be an absolutely convergent series, and be a conditionally convergent series. Then any rearrangement of is convergent to the same sum. It is a result due to Riemann that can be rearranged to converge to any sum, or not converge at all.
|Date of creation||2013-03-22 12:24:51|
|Last modified on||2013-03-22 12:24:51|
|Last modified by||yark (2760)|