22 ms·
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 s
by gsk 15y ago
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.
- Someone 15y ago"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.
- gjm11 15y agoI 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.)
- deleted 15y ago[deleted]