Associated Topics || Dr. Math Home || Search Dr. Math

### Summing an Oscillating Series

```
Date: 08/10/98 at 22:42:00
From: Anonymous
Subject: Associative property in infinite series.

In a debate, I said that (1-1) + (1-1) + (1-1) ... does not equal
1 + (-1 + 1) + (-1 + 1) + (-1 + 1) ....

This is what I wrote:

0 + 0 + 0 ... = (1-1) + (1-1) + (1-1) ...

And by associative property, I said:

Then 0 + 0 + 0... = 1 + (-1+1) + (-1+1) + (-1+1)...
And thus, 1 = 0.

I took a part of an infinite series that repeats itself, and I simply
broke the pattern by adding an extra one to it. I'm having trouble
expressing my thoughts, but I'll do my best:

Let (1-1) + (1-1) + (1-1)... = a
By commutative property, (-1+1) = 1-1

Now do you see the picture? The expression I made is actually 1 + a.
And if you think about it again, infinite series doesn't just go on in
one direction. It goes in both directions. You can't really have one
end of the series showing because there is always something in front of
it. Thus, what you have is an expresson from the middle of the series.
So if you want to properly follow the associative property of addition,
you should have the first one in parentheses with a negative one.
Otherwise, you are simply adding one to zero.

I would like to know if there is something wrong with my argument or
if I could add something to strengthen my point. Any comment will be
appreciated.
```

```
Date: 08/11/98 at 01:22:45
From: Doctor Schwa
Subject: Re: Associative property in infinite series.

The standard mathematician's answer would be that this series does not
converge - that is, it does not have a well-defined sum, for precisely
the reason you give. It oscillates between 1 and 0. The usual
definition of the sum of a series is the limit of s_n, where s_n is the
sum of the first n terms of the series. In this case, where the
sequence s_n runs 1, 0, 1, 0, 1, 0, ..., the limit doesn't exist.

On the other hand, series like this do sometimes come up in physics.
And there are alternate definitions of summation of infinite series
that you can use. In some of those, the sum of this series is well-
defined, and in fact is equal to 1/2.

So what do you think the sum of 1 - 2 + 3 - 4 + 5 - 6 + 7 ...
should be?

- Doctor Schwa, The Math Forum
Check out our web site! http://mathforum.org/dr.math/
```
Associated Topics:
High School Physics/Chemistry
High School Sequences, Series

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search