# The series ∑∞n=11n\sum_{n=1}^\infty\frac1n diverges!

We all know that the following harmonic series

diverges and grows very slowly!! I have seen many proofs of the result but recently found the following:
In this way we see that $S > S$.

Can we conclude from this that $S$ is divergent??

The proof can be made a bit more rigorous by setting

Note that $a_n\ge b_n$, $a_n\gt b_n$ when $n$ is odd, and $a_n=b_{2n-1}+b_{2n}$.

Assuming that

converges, then

also converges. However,

Since $a_n\ge b_n$ and $a_n\gt b_n$ when $n$ is odd.

Now, $(3)$ says that

and $(4)$ says that

These last two statements are contradictory, so the assumption that $(2)$ converges must be false.