Is there an intuitive non-mathematical way to show $\sum_n 1/n=\infty$?

Question

I want to show some school students about the sum of the harmonic series diverging and I need some nice interpretation for this. I'll preferably love to have a figure.

Thanks in advance!

Answer 1

One way is to bound the series below by a series that more immediately looks infinite. We round down $1/n$ to largest fraction of the form $1/2^k,k\in\mathbb N$. This is illustrated as such: \begin{align*} &1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\ldots\\ \geq&1+\frac{1}{2}+\frac{1}{4}+\frac{1}{4}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}\ldots \end{align*} Hopefully that presentation is clear enough. We may combine terms in this lower sum to see (loosely) that we are adding up $1/2$ an infinite number of times. This shows that the sum diverges, at least sufficiently for illustrative purposes.