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:
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