Way Back to Newton

One Reply to “Way Back to Newton”

1. I love power series, both because they are a powerful tool and because people like Euler used them in such imaginative and playful ways.

   As an imaginative and playful example, think about history’s first power series, the geometric series. It’s got a formula that people might have learned in school:
   $$
   1+x+x^2+x^3+\cdots=\frac1{1-x}.
   $$
   It’s easy to find a proof of this online.

   Since the discovery/invention of limits in the 19th century, we’ve been careful to tell students that this formula can only be used if the series converges (for the cognoesenti, this happens iff the partial sums have a limit, which happens iff $-1<x<1$). In Euler's unlimited world, though, his world without limits, the formula was used regardless of the value of $x$. So Euler was perfectly happy to let $x=-1$ and to say
   \begin{align*}
   &1+(-1)+(-1)^2+(-1)^3+(-1)^4+\cdots\\
   &\qquad=1-1+1-1+1-1+\cdots\\
   &\qquad=\frac1{1+1}=\frac12.
   \end{align*}
   You add up a bunch of whole numbers, and you get a fraction.

   Euler was even willing to let $x=2$ and to say
   \begin{align*}
   1+2^1+2^2+2^3+2^4+\cdots&=\\
   1+2+4+8+16\dots&=\frac1{1+(-2)}=-1.
   \end{align*}
   You add up a bunch of positive numbers, and you get something negative. From a modern point of view, this seems mildly nuts, but Euler wasn't bothered by the idea that the numbers form a sort of ring where $+\infty$ and $-\infty$ meet. In fact, although both these calculations are clearly wrong if you believe the definition of infinite sums you get taught in calculus classes, there are modern perspectives (look up Cesàro summability and $p$-adic numbers) from which they are deep and insightful and correct.

   A final bit of wild imagination applied to a geometric series: In 1703, Guido Grandi did this:
   \begin{align*}
   0&= 0+0+0+0+\cdots\\
   &=(1-1)+(1-1)+(1-1)+(1-1)+\cdots\\
   &=1-1+1-1+1-1+1-1+\cdots\\
   &=1+(-1+1)+(-1+1)+(-1+1)+\cdots\\
   &=1+0+0+0+\cdots\\&=1.
   \end{align*}
   Something has been made from nothing, Grandi says, so should we be surprised that the world was made by God out of nothing?

   When I first met this argument by Grandi, I thought he must have been a wonderfully sweet man to find God in a divergent series, and I hoped I might meet him after my death. Sadly, what I have read about him subsequently seems to say that he was quite irritating and argumentative. I'm sad, but we don't always get what we want.

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