Comment by Someone

There’s a trivial lower bound of zero, for n = 1.

For n > 1, there isn’t a lower bound. None of the numbers are integers again (https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)#...), and because the difference between successive partial sums goes to zero and the series grows to arbitrary values, you’re bound to find a difference smaller than 1/(2n) somewhere beyond n.