Have you tried dividing 987654321 by 123456789? If not, do it, the result is astonishing.

Here’s why the result isn’t actually that astonishing.

We start with the sum of a geometric series (valid for |x|<1

∑∞n=0xn=11−x

Differentiating both sides gives

∑∞n=1nxn−1=1(1−x)2

and hence

∑∞n=1nxn=x(1−x)2

Going back to the original series, subtracting the first term gives

∑∞n=1xn=11−x−1=x1−x

If k

is some constant, we have

∑∞n=1kxn=kx1−x

and hence

∑∞n=1(k−n)xn=kx1−x−x(1−x)2=kx(1−x)−x(1−x)2
 

The ratio between these is

∑∞n=1(k−n)xn∑∞n=1nxn=kx(1−x)−xx=k(1−x)−1


Now, when x=110
and k=10

this becomes

∑∞n=110−n10n∑∞n=1n10n=10(910)−1=8

But a number expressed in decimal is just a particular kind of sum involving powers of ten:

0.123456789=1101+2102+⋯+9109=∑9n=1n10n

and, likewise,

0.987654321=∑9n=110−n10n

that is, these numbers are just the sums in the numerator and denominator, but truncated after 9
terms. That means that their ratio should be approximately 8, with a small amount of “error” resulting from truncating the sums. Nothing astonishing; just algebra and calculus.

Source: Some or all of the content was generated using an AI language model