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.
