Okay so had another look at this.

So to prove the sum is not an integer.

So,

Let

An overestimate for the sum would be

An underestimate for the sum would be

But since

Thus the sum is not an integer.

Use induction to prove it is a zero of a monic polynomial.

Base case:

Let the sum be written as

Let

.

Thus

Thus

is a zero.

Now for the inductive hypothesis, assume

is true.

Thus

is true.

Now we wish to prove that

is also true.

We will denote this as

.

Let this be a zero of a monic polynomial

So using the inductive hypothesis,

is a zero

, namely

so

Now what...?