stoutfellow

If there is one thing Leonhard Euler is better known for than anything else, it is his virtuoso work with infinite series. That he came up with more ways of manipulating them than anyone before or since is not disputed. That his work was less than rigorous by modern standards is also not disputed. Both of these judgments, though, fall short of the full truth, and the next two or three Rambles will discuss the matter. More below the cut.

First, I want to take a look at some fundamentals. At some point, the reader may have run across the following argument, that the infinite decimal 0.3333... is equal to 1/3.

Let x=0.3333.... Then 10x=3.3333....=3+x. Subtracting x from both sides, we obtain 9x=3, or x=1/3.

It is not quite obvious, but what we're dealing with here is an infinite series. 0.3333... is shorthand for the series 0.3+0.03+0.003+0.0003+..., in which each term is ten times the next. Thus, if we multiply by 10, the series becomes 3+0.3+0.03+0.003+..., and the argument proceeds as above. So far, so good; but allow me to present two more arguments in the same vein.

Let x=1+2+4+8+16+..., in which each term is twice the one before. Then 2x=2+4+8+16+32+...=x-1, from which we deduce that x=-1.

Again:

Let x=1-1+1-1+1-..., in which the terms alternate between 1 and -1. Then -x=-1+1-1+1-1+...=x-1, from which we see that x=1/2.

The second and third arguments are plainly absurd; yet they have exactly the same shape as the first - and the first argument seems unexceptionable. Something is going on; some extra element must be involved, which was not mentioned. If you've taken enough calculus, you may believe you know what the explanation is. The truth, though, is more complex than you were told - but that's matter for another post.

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