Changing the Order of Summation
The terms of a finite sum can be added in any order without changing the result, since
addition is commutative. The same is not always true for infinite series! This seems
very strange, and, if it is true, makes infinite series seem very wierd, pathological
objects. The good news is that the the scope of this problem can be completely understood.
There is one case where everything that can go wrong, does, and in the other case, there is
no problem with adding the terms of the series in any order. The division between the
well-behaved and the wild series is the division between absolutely and conditionally
First of all, let's decide what we mean by changing the order of addition of the terms of a series. If we add four real numbers
then there are 4!=24 ways of writing the terms in order, including the possibility of reversing the terms:
It doesn't seem reasonable to expect to reverse the order of the terms in an infinite series---there is no 'last' term to put first!
There's now a very nice split: Every conditionally convergent series can be rearranged so as to diverges, or even, to converge to any real value at all. In stark contrast, every absolutely convergent series can be rearranged freely, without affecting its convergence, or the value that it converges to in the least! The next two propositions set out the details of these facts.
The idea of the proof.
Try out an interactive demo of this construction
Of course it takes quite a delicate argument to make a rearrangement of a conditionally convergent series converge to a given value. In general you shouldn't expect a rearrangement of a conditionally convergent series to converge at all. A conditionally convergent series can always be rearranged so as to diverge.
Analysis WebNotes by John Lindsay Orr.
Comments to the author: firstname.lastname@example.org
All contents copyright (C) 1996 John L. Orr University of Nebraska--Lincoln All rights reserved
Last modified: May 1996