Remember from the last post that the sum of inverses of all strictly positive integers converges to infinity:

I also proved that the sum of inverses of all strictly positive integers that don’t contain the digit 9 is finite. Denote this sum .

Let be the sum of all strictly positive integers that contain the digit 9:

Then:

Thus , and it follows that because and is finite.

In words: the sum of inverses of strictly positive integers that don’t contain digit 9 is finite and the sum of inverses of strictly positive integers that contain digit 9 is infinity.

This may be counterintuitive because apparently there are more numbers that don’t contain digit 9 than numbers that contain digit 9, but this is not actually true.

There are strictly positive integers up to digits that don’t contain digit 9 (see here). And obviously the total number of strictly positive integers is . This means that the number of strictly positive integers that contain digit 9 is: . And here’s the trick, for n large enough: .

Let’s plot . You can see that for (horizontal axis) is much bigger than .

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