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

is never an integer. This was proved by Jozsef Kurschak in 1908.

This means that the harmonic numbers defined by

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.

Kurschak'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, H_n - H_m is never an integer unless m = n.

Related posts

• Computing harmonic numbers
• Numerators of harmonic numbers
• Summing floating point numbers
• Special numbers

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