## Harmonic series

Posted December 9th, 2007 by nico

We want to prove that harmonic series is divergent.

One elementary solution is an easy consequence of some well known inequality

This is equivalent to

or

This inequality is also a consequence of Lagrange theorem applied to logarithmic function on interval [n,n+1].

Summing up from 1 to n we get

or

From the right side of the above relation we have

Now let consider harmonic series

As

taking limit in both sides we also have

As the partial sum of the harmonic series is a divergent sequence, the harmonic series is divergent.