(a downloadable pdf version of the article is available here )
In the previous article I showed that the Harmonic Series diverges. It implies that given an integer k, we can truncate the harmonic series at a sufficiently large n to have the truncated sum larger than k. An obvious question that arises here is -- is this truncated series (called as the nth Harmonic number) ever an integer.
For n = 1, we see that obviously it is indeed an integer. We ask whether this is true for any other n? The following theorem shows that this is not possible. That we can never have the nth harmonic number equal to an integer for any n.
Theorem: The nth Harmonic Number cannot be an integer for any n>1.
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.