From your earliest days of math you learned that
the order in which you add two numbers doesn't matter:
3+5 and 5+3 give the same result. The same is true
for the addition of any finite set of numbers.
But what if you are adding an infinite set of numbers?
For instance, in calculus
you encounter series, which are infinite sums.
Does the order of addition matter?
Surprisingly, yes, but only for some series.
A series is said to converge absolutely if the sum
of the absolute values of the terms converges. It is
a fact that a series which converges absolutely
must converge, and no matter how you rearrange the terms
the sum will be the same.
Series that do not converge absolutely are wilder;
they may or may not converge (so they are called
conditionally convergent). But, if such
a sequence does converge, it is even stranger still
by rearranging the terms in the sequence, you can get
the series to converge to any value you want!
Presentation Suggestions:
Do an example with the alternating harmonic series:
1  (1/2) + (1/3)  (1/4) + ...
This series is conditionally convergent
(because the
harmonic series
diverges).
But by the alternating series test,
it convergesin fact, to ln(2).
As a demonstration, show students how to rearrange it to
converge
to any value (see Math Behind the Fact).
The Math Behind the Fact:
This fact is due to Riemann, and the basic idea is simple.
To obtain a rearrangement that sums to X, just add enough
positive terms from the series (in their original order)
until the partial sums first exceed X, then
add enough negative terms from the series
(in their original order) until the partial sums first
dip below X,
then repeat and continue with the successive positive terms,
then the successive negative terms, etc.
This will yield a rearrangement of the original series
that sums to X.
This fact shows how careful one has
to be when switching the order of addition in a
conditionally convergent series.
How to Cite this Page:
Su, Francis E., et al. "Does Order of Addition Matter?."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
