Formal Sums
Is it possible to assign a unique value to every infinite sum in a way that is
consistent with the axioms of the real numbers?
This question will no doubt elicit diverse responses .
For example, one could
answer in the affirmative by assigning the value zero to every infinite sum.
There is no inherent contradiction in making such an assignment.
But
obviously, such a definition trivializes the potential of infinite processes as a tool
of analysis.
Another common answer is negative, due to problems that arise with divergent
sums.
An example would be the following:
Suppose we assign a value , say
x
, to the sum 1 + 1 + 1 + 1 .
.. then we could
write
x
= 1 + 1 + 1 +
... =
1 + ( 1 + 1 + 1 + .
.. ), or
x
= 1 +
x
.
Subtracting
x
from
both sides, we find that 0 = 1, a contradiction.
Consequently, this divergent sum
cannot be assigned any finite value consistent with the axioms of arithmetic.
This example bolsters a commonly held belief that certain infinite sums are, by
their very nature, outside the realm of real arithmetic.
If we look at the partial
sums of the above example:
1=1
1+1 = 2
1+1+1 = 3
1+1+1+
...
+1
=
n
it seems evident that
x
has a value that is greater than any finite number.
This
judgement is reinforced by the equation
x
= 1 +
x
which could reasonably be
construed to have the solution "
∞
".
Then, by labelling operations such as
"
∞

Author: Chappell Brown
First created:
September , 1992
Last revision: April, 1996
Page:
1
© Copyright 1996 All rights reserved. Permission to copy and distribute for nonprofit scholarly purposes granted.
http://www.limit.com
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentFormal Sums
∞
" as undefined, we
avoid the contradictory result 0 = 1 .
This common textbook example seems to force us to the conclusion that
mathematicians arrived at in the last century, that infinite sums by their nature
divide into two classes: convergent sums for which meaningful finite values
exists, and divergent sums, for which no reasonable value can be found.
As
compelling as these, and other examples like them seem, they do not actually
constitute a rigorous proof of this belief.
Indeed,
an examination of our
introductory question will reveal that the dichotomy between convergent and
divergent processes is more an article of faith than an artifact of logic.
How exactly do we get at this question in a comprehensive and rigorous fashion?
The intellectual development of the calculus over the past four centuries could
be viewed as an experiment to nail down a definitive answer.
Newton
introduced infinite series as a tool for studying the behavior of complex processes
in terms of simple and well understood functions.
A statement such as:
e
x
=
1
+ x
+
x
2
2!
+
x
3
3!
+
was viewed as a means for characterizing
a complex function such as e
x
in
terms of a series of simpler polynomials.
Newton and his contemporaries found
that the smaller the value of x, the more quickly the series of polynomials:
1 ,
1
+ x
,
1
+ x
+
x
2
2!
,
1
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '04
 IgorDolgachev
 Real Numbers, Mathematical analysis, Chappell Brown, nonprofit scholarly purposes, Formal Sums

Click to edit the document details