Prove that 1/1 + 1/2 + 1/3 + ............... + 1/n is not an integer when n is a positive integer and n>1.

Theorem: The nth Harmonic Number cannot be an integer for any n>1.

Proof:
Let n >1. We denote the nth partial sum by

We can write the nth partial sum as a summation of n terms in the numerator each of the form where 1≤k≤n. The denominator is

So the nth partial sum can then be written as

Now the strategy is to show that the highest power of any prime p dividing the denominator is strictly greater than that dividing the numerator.

We choose p=2 so that our proof will work for all
It is obvious that there exists a r such that

Let the highest power of 2 that divides the denominator be t th power.

Now in each of the terms in the numerator the whole product of the first n natural numbers is divided by an integer k. Note that if this k has the highest power of 2 dividing it as x then the highest power of 2 dividing that particular term will be t-x.
Note that the maximum value of x can be r and hence the minimum value of (t-x) can be (t-r).

This is attained for Note also that every other term in the numerator has the highest power of 2 dividing it , strictly greater than (t-r). This is because every other k has the highest power of 2 dividing it strictly less than r.

Thus all the terms in the numerator for have dividing them while the term corresponding to k=2^r cannot have it.

Thus the highest power of 2 dividing the numerator is equal to (t-r).

For n≥2 ...r≥1 and hence the highest power of 2 dividing the numerator is strictly less than the highest power of 2 dividing the denominator.

This implies that the nth partial sum can never be an integer for any n≥2.