Another interesting one is that if you remove everything but the prime numbers, it still diverges. So there are more primes than numbers without a 9 in them.
Given that prime frequency is around ln(n) on average, and that nineless frequency is around .9^log_10(n) (log-exponentially low, or whatever the right terminology is), this isn't actually that surprising when you think about it.
It's only 0 as far as statisticians and cheap calculators are concerned. The number of N-digit numbers without 9s in them is much higher than the number of similar (N-1)-digit numbers, it's just the proportion that falls. The probability isn't 0, it just asymptotically approaches 0, as the number of non-9-containing numbers approaches infinity.
In essence, the cardinality of the set of all numbers that do not contain 9 is infinity; the cardinality of the set of all numbers is infinitier. This despite the fact that both sets are countably infinite.
So, this is just a parlor trick, right? It sounds like this is just a complicated way of saying that this is what the harmonic series looks like when you've removed all terms with more than N digits.
Not really, because you've removed 'almost all' terms, but there are many ways to remove 'almost all' terms that do not result in a convergent series. For instance, you could remove 999 out of every 1000 terms. I'd say that counts as removing 'almost all' terms. Nevertheless, the resulting series does not converge, for the exact same reason that removing every other term doesn't work. The amount of terms that you remove has to increase faster than that.
And in particular, removing 100% of the terms doesn't mean much either, since, for example, the series 1/p for all primes does not converge, or 1/floor(n log n). You need to remove them harder than that.
You can remove 8, 7 , 1 ,2... in fact, any digit and that will make the series convergent. You can go further. Remove 'any' group of digits that occurs in the series repeatedly (say, all terms with 373737 somewhere in it) and the series will converge. This is because as the terms get bigger, it becomes increasingly common to find your chosen 'digit' in almost all the terms, thus making the series convergent.
"as the terms get bigger, it becomes increasingly common to find your chosen 'digit' in almost all the terms, thus making the series convergent."
No! I think that condition is necessary, but it certainly is not sufficient for the series to converge. For example, non-primes become increasingly common, but the sum of their reciprocals is divergent.
I don't think gsk was claiming it was a sufficient condition -- just trying to give some intuition for why the phenomenon happens. (After all, if you only look at very small numbers is seems like most don't have 9s.)
