Why does the series $\sum_{n=1}^\infty\frac1n$ not converge? Can someone give a simple explanation as to why the harmonic series 

$$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$

doesn't converge, on the other hand it grows very slowly? 
I'd prefer an easily comprehensible explanation rather than a rigorous proof regularly found in undergraduate textbooks.

The answer given by AgCl is a classic one. And possibly pedagogically best; I don't know.
I also like the following argument. I'm not sure what students who are new to the topic will think about it.
Suppose 1 + 1/2 + 1/3 + 1/4 + ... adds up to some finite total S. Now group terms in the following way:
$$1 + \frac{1}{2} > \frac{1}{2} + \frac{1}{2} = \frac{2}{2} = 1$$
$$\frac{1}{3} + \frac{1}{4} > \frac{1}{4} + \frac{1}{4} = \frac{2}{4} = \frac{1}{2}$$
$$\frac{1}{5} + \frac{1}{6} > \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$$
Continuing in this way, we get $S > S$, a contradiction.
