There are a couple ways to prove this that I know of. Your title says number theory so if you have discussed in class (or read on your own) p-adic valuations, try to show that for a fixed n>1 the 2-adic valuation is always greater than 1. If you don't have experience with this, try the sum where n=p (some prime). Make an argument (manipulate terms) why this isnt' an integer. Then fix that p, and pick an x between p and 2p-1, and follow a similar argument. Unfortunately you now must prove that for any x such a p exists. I believe this was proven by Erdos. Maybe your teacher will let you get away with that as justification. Another way is to show that for some n, the sum=(even number + odd number)/2N *odd number which isn't an integer.