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.

This is not as good an answer as AgCl's, nonetheless people may find it interesting.
If you're used to calculus then you might notice that the sum $$
1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n}$$ is very close to the integral from $1$ to $n$ of $\frac{1}{x}$.  This definite integral is ln(n), so you should expect $1+\frac{1}{2}+\frac{1}{3}+\dots+
\frac{1}{n}$ to grow like $\ln(n)$.
Although this argument can be made rigorous, it's still unsatisfying because it depends on the fact that the derivative of $\ln(x)$ is $\frac{1}{x}$, which is probably harder than the original question.  Nonetheless it does illustrate a good general heuristic for quickly determining how sums behave if you already know calculus.
