Problem statement here.
The problem was to show that fractional parts of harmonic numbers can get arbitrarily close to zero.
It is known that harmonic numbers are divergent. A quick proof of this can be done using the fact that 1x is the derivate of logx and applying Rolle's theorem to get:
1n+1<log(n+1)−logn<1n
Add up and get that harmonic number Hn is at least as big as log(n+1) and so diverges.
Now let nk be the smallest integer such that Hnk>k.
We must have that Hnk−1<k<Hnk i.e
Hnk−1nk<k<Hnk
Thus the fractional part of Hnk is no more than 1nk. Since nk→∞ as k→∞, the fractional parts can be made arbitrarily small.
nk grows approximately as ek−γ where γ is the euler mascheroni constant. It is known that nk is either ⌊ek−γ⌋ or ⌊ek−γ⌋+1.
It is also conjectured to be the integer nearest to ek−γ+12, attributed to Comtet (who proved the statement in the previous paragraph).
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