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[Solution] Fractional parts of Harmonic numbers


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 Hnk1<k<Hnk i.e

Hnk1nk<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 ekγ where γ is the euler mascheroni constant. It is known that nk is either ekγ or  ekγ+1.

It is also conjectured to be the integer nearest to ekγ+12, attributed to Comtet (who proved the statement in the previous paragraph).

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