The harmonic series, after removing any term with a "9" in it, is convergent.

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.

Well of course, the vast majority of large numbers have nines in them. All but 0% of them.

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.

There is a discussion of this over at /r/math: <a href="http://www.reddit.com/r/math/comments/if4kd/take_the_harmonic_series_remove_any_term_with_a_9/" rel="nofollow">http://www.reddit.com/r/math/comments/if4kd/take_the_harmoni...</a>