By 'easy' I mean avoiding heavy notation. Ideally we only need numbers and multiplication signs. You can use algebra implicitly, but try to keep things elementary school level.
Two examples of what I mean:
The sum of reciprocals is infinite: The first term is $1$. Take the next 9 terms. The smallest one is $\frac{1}{10}$ so their sum is at least $\frac{9}{10}$. That makes 10 terms. Take the next 90 terms. The smallest one is $\frac{1}{100}$ so their sum is at least $\frac{90}{100} = \frac{9}{10}$. That makes 100 terms. Take the next 900 terms. The smallest one is $\frac{900}{1000} = \frac{9}{10}$. Proceeding in this manner, we see the sum is greater than $1+\frac{9}{10}+\frac{9}{10}+\ldots +\frac{9}{10}$ for any number of $+\frac{9}{10}$s.
The sum of powers of halves is finite: Eat half a cookie. Now eat half of whats left. Now eat half of what's left. Now eat half of what's left. The remainder halves every time. So after eating forever you eat the whole cookie. So $\frac12+\frac14+\frac18+ \ldots = 1$.
Is there a similar trick for the series $\sum_{n=1}^\infty \frac{1}{n^2}$?
