has a finite sum , then the addition in (1) can be performed in reverse , i.e.
Proof. The assumption on (2) implies that the sum of an arbitrary finite amount of the numbers is always . This means that (1) is absolutely convergent, and thus the order of summing is insignificant.
Note. The series satisfying the assumptions of the theorem is often denoted by
and this may by interpreted to an arbitrary summing . One can use e.g. the diagonal summing:
|Date of creation||2013-03-22 16:32:54|
|Last modified on||2013-03-22 16:32:54|
|Last modified by||PrimeFan (13766)|
|Synonym||double series theorem|