# No rational function

Prove that there exists no rational function such that

Solution

Suppose , on the contrary , that such function exists. Since the harmonic series diverges we conclude that the limit of our function in infinity is infinity. This, in return means that the degree of the nominator , call that is greater that the one of the denominator , call that . Extracting in the nominator we get that

The limit of at infinity is finite and call that . Hence:

and of course this contradicts the fact that

where is the Euler  – Mascheroni constant .