# Finite Sum & Divisibility

Standard

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

$\displaystyle{\sum_{k=2}^{n} \frac{1}{k}}$  can never be an  integer for any value of $n$.

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 $n$ 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 $n$. And the numerator will be an odd number since it’s the sum of $(n-2)$ 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.

# [poem] Harmonic Noise

Standard

———-
Harmonic Noise

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.

-Gaurish
————
For more details about Harmonic series refer Wikipedia: https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)