Sums of consecutive reciprocals

The sum of the reciprocals of consecutive integers is never an integer. That is, for all positive integers m and n with n > m, the sum

\sum_{k=m}^n \frac{1}{k}

is never an integer. This was proved by József Kürschák in 1908.

This means that the harmonic numbers defined by

H_n = \sum{k=1}^n \frac{1}{k}

are never integers for n > 1. The harmonic series diverges, so the sequence of harmonic numbers goes off to infinity, but it does so carefully avoiding all integers along the way.

Kürschák’s theorem says that not only are the harmonic numbers never integers, the difference of two distinct harmonic numbers is never an integer. That is, HnHm is never an integer unless m = n.

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