Wednesday, March 01, 2006

An Infinite Series With A Precise, Well Defined Sum

Suppose that we have the infinite series 1, 1/16, -1/512, 1/8192… and we wish to take its sum (that is, we wish to add all the numbers in the sequence together, up to whatever number happens to occupy the infinite position in the sequence. Those numbers might look random, but, as I will show later, they are produced by a definite mathematical rule such that any term whatsoever in the sequence can be produced.

I can guarantee you that the sum of the terms of the sequence I just gave is 3/sqrt(8) – that is, 3 divided by the square root of 8. How can I do that? The sequence of numbers that I just gave happens to be the binomial expansion for (1+1/8)^(1/2) – in other words, the binomial expansion of the square root of 1+1/8. Of course, if you multiply that square root by the square root of 8, you get the square root of 1+8, which is the square root of 9. As we all already know, the square root of 9 is 3. Thus, if you really believe that the square root of 9 is 3, then you must also accept that the sum of the sequence I just gave is 3/sqrt(8).

In the case that you don’t believe me that the binomial expansion can yield a definite answer, or that the binomial expansion is actually correct, I suggest you check out this web site (which contains a small proof):

The same arguments which I just presented can be made, in a similar form, for many other series and sequences.

Also, please note that it is possible find the exact value of a definite integral. For those of you who know anything at all about calculus, it should be readily apparent why that’s important in this context.

Also, note that for any system of logic which may be constructed, there is a corresponding system of mathematics. Therefore, we are left with the inescapable conclusion that if our logic is sound, then there is a sound mathematical theorem to go along with it.


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