In this short post we are going to study the sum where is a permutation of . As you can see, this a variation of the famous harmonic sum whose value is equal to .

If you want to explore this problem by yourself before I reveal more, now is the time. I do believe that fiddling with this sum before reading on would make the rest of this blogpost more interesting.

The inequality

We are going to prove that

Again, if you want to find a proof by yourself, now is the time.

Proof number 1

We are going to use Cauchy-Schwarz inequality. To do so, consider the infinite vectors and , and note that their scalar product is exactly . By Cauchy-Schwarz inequality we have And since we can directly deduce the inequality we wanted to prove.

Proof number 2

For this proof, we are going to use the AM-GM inequality. For every we have that and thus, by the AM-GM inequality, we obtain By summing all these terms for we obtain and since we can directly deduce the inequality we wanted to prove.