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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 

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
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Associated Topics:
High School Physics/Chemistry
High School Sequences, Series

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