Summing an Oscillating SeriesDate: 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/ |
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