I wish to discuss a small problem from The USSR Olympiad Problem Book (problem 59) about the finite sum of harmonic series. The problem asks us to prove that
can never be an integer for any value of .
I myself couldn’t think much about how to prove such a statement. So by reading the solution, I realised that how a simple observation about parity leads to this conclusion.
Firstly, observe that among the natural numbers from 2 to there is exactly one natural number which has the highest power of 2 as its divisor. Now, while summing up the reciprocals of these natural numbers we will get a fraction as the answer. In that fraction, the denominator will be an even number since it’s the least common multiple of all numbers from 2 to . And the numerator will be an odd number since it’s the sum of even numbers with one odd number (corresponding to the reciprocal of the number with the highest power of 2 as the factor). Since under no circumstances an even number can completely divide an odd number, denominator can’t be a factor of the numerator. Hence the fraction can’t be reduced to an integer and the sum can never be an integer.
One day I heard something musical,
It sounded to me mystical.
Since it was ordered,
A harmonic pattern was being followed.
The pattern was elegant,
It was inverse of each natural number element.
I made the number elements bigger hero,
The pattern approached symmetrical zero.
When I summed up the pattern without bound,
I got only noise without any musical sound.
For more details about Harmonic series refer Wikipedia: https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)
Edit (20 May 2018): A video by Marc Chamberland discussing a puzzle involving harmonic series.