The Sum:

Does not yield an integer for all .

For k = 2:

That is not an integer.

Since the series is divergent, then we must reduce it:

To have an integer it has to be that for some integer M:

This can be rewritten as:

We can say that:

So that:

We can also say that:

So that our final equation is:

The series is known as the Harmonic series, and is often dealt with for primes.

If k is prime, this is the sum of a multiple and a non-multiple of k.

Clearly there is no M that satisfies the equation.

The Harmonic Series does not give an integer for any