TITLE: Alternative ways to show that the Harmonic series diverges
QUESTION [7 upvotes]: This question came to me from one of my calculus students today: Other than using the integral test $$\int_1^\infty \frac{dx}{x} \to \infty,$$ what are some other ways that we can prove the Harmonic series $\sum_{n=1}^\infty \frac{1}{n}$ diverges?
I'm sure there are plenty of methods out there; any method where a typical student in Calculus could understand would be great.

REPLY [2 votes]: Suppose
$$S=1+{1\over2}+{1\over3}+{1\over4}+\cdots$$
is convergent.  Since all the terms are non-negative, it must be absolutely convergent.  But absolutely convergent sequences can be fiddled with at will, allowing us to conclude
$$\begin{align}
2S&=2+1+{2\over3}+{1\over2}+\cdots\\
&=S+\left(2+{2\over3}+{2\over5}+\cdots\right)\\
&\gt S+\left(1+{1\over2}+{1\over3}+\cdots\right)\\
&=2S
\end{align}$$
which is absurd:  No number is greater than itself.