Ramble, Part 38: The Sum of Things
Nov. 10th, 2007 05:31 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowThere remains the third series, 1-1+1-1+..., for which I gave a (fallacious) argument that it equalled 1/2. Unlike the preceding example, this one actually does go back to Euler; details are under the cut.
Euler was perfectly aware that the series 1-1+1-1+... does not have a sum, using the standard definition. The partial sums 1, 1-1, 1-1+1, 1-1+1-1, etc, alternate between 1 and 0; they do not close in on any specific value. (The trick in the last post, of changing the definition of distance, isn't going to work in a situation like this.) Nonetheless, he held to the claim that it should be regarded as summing to 1/2.
Let me first point out that there is nothing inherently wrong with this. As I said earlier, it is impossible to literally add up infinitely many numbers; the best we can do is come up with something analogous, and different analogies will yield different results, which may be of value in different ways. Euler's argument was, more or less, that the partial sums, being half 1's and half 0's, average to 1/2. Though he did not pursue this with great rigor, it is possible to make a solid definition on this basis. A series is "Cesàro summable" if the running average of the partial sums converges. In this case, the averages are 1, 1/2, 2/3, 1/2, 3/5, ..., and they do indeed converge to 1/2.
Why is this of any interest? Well... there is a theorem, of rather a technical nature, which justifies it. Every Cesàro summable series turns out to be "Abel summable"; loosely speaking, there is a sequence of series whose terms converge to the terms of the series; those series themselves have sums in the usual sense, and their sums converge to the Cesàro sum of the original series. I'll blockquote the details, so the less mathematically inclined can skip over them.
There's a lesson in there, which will become clearer when we proceed into the nineteenth century; but there are a couple of other items to be discussed before we get there.
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Euler was perfectly aware that the series 1-1+1-1+... does not have a sum, using the standard definition. The partial sums 1, 1-1, 1-1+1, 1-1+1-1, etc, alternate between 1 and 0; they do not close in on any specific value. (The trick in the last post, of changing the definition of distance, isn't going to work in a situation like this.) Nonetheless, he held to the claim that it should be regarded as summing to 1/2.
Let me first point out that there is nothing inherently wrong with this. As I said earlier, it is impossible to literally add up infinitely many numbers; the best we can do is come up with something analogous, and different analogies will yield different results, which may be of value in different ways. Euler's argument was, more or less, that the partial sums, being half 1's and half 0's, average to 1/2. Though he did not pursue this with great rigor, it is possible to make a solid definition on this basis. A series is "Cesàro summable" if the running average of the partial sums converges. In this case, the averages are 1, 1/2, 2/3, 1/2, 3/5, ..., and they do indeed converge to 1/2.
Why is this of any interest? Well... there is a theorem, of rather a technical nature, which justifies it. Every Cesàro summable series turns out to be "Abel summable"; loosely speaking, there is a sequence of series whose terms converge to the terms of the series; those series themselves have sums in the usual sense, and their sums converge to the Cesàro sum of the original series. I'll blockquote the details, so the less mathematically inclined can skip over them.
Suppose our Cesàro summable series has terms a0, a1, a2, .... Consider the power series a0+a1x+a2x2+.... If this series converges for every choice of x between 0 and 1, and if the values it takes converge as x approaches 1, then we call the series "Abel summable". In this case, we're looking at the series 1-x+x2-x3+..., which is well-known to converge to 1/(1+x) if |x| is less than 1; the limit of this last expression as x approaches 1 is, of course, 1/2.The point, in any case, is that every way of adding infinitely many numbers is more or less arbitrary; the real question is whether any given way of doing so is useful, and under what circumstances. Euler's manipulations of infinite sums (and infinite products as well) frequently were not rigorous by latter-day standards, but usually there was some justification for them, and appropriate alternative definitions could be found which made them perfectly acceptable.
There's a lesson in there, which will become clearer when we proceed into the nineteenth century; but there are a couple of other items to be discussed before we get there.
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