read any fraction hereunder as:
(quantity of digits used by the last term) ______________________________________________
(quantity of digits used in the sequence so far)
Does the sum of all terms converge towards a precise number?
[spoiler space ...] ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... [... spoiler space] Assuming for the moment that you don't reduce fractions to their lowest terms: no, the sum diverges, because the terms with numerator n are (more or less) evenly spread over 10^(n-1)..10^n with density 1/n, giving a total of about log 10, which doesn't -> 0. If you do reduce to lowest terms, then the result is (at most) that you use fewer digits, so the numbers grow more slowly, so the fractions shrink more slowly. If you ever do get cancellation, that is; it appears that you don't for the first few decades, but I'm too lazy (and possibly too stupid; I can't tell because I'm too lazy) to work out whether you ever do. -- g