# A proof of the divergence of the harmonic series

Alexander Frolkin

8 August 2001

We start off with the following definition of the Riemann Zeta function:

We turn our attention to the integrand:

Some manipulation of the series yields

Plugging back into the Zeta function,

Substitute in the integral1, so that

Plugging back into the definition,

This leads to the more common definition of the Riemann Zeta function,

If we put , the function reduces to the harmonic series --

We can find the value of using the integral definition. So, we have

To evaluate the integral

substitute

Hence,

Since these are improper integrals, in the Riemann sense, we need to take limits. Now,

We have

Since

We have

Hence,

completing the proof that the harmonic series diverges.