%% I wish I had a way to make all the figures old-style, but Plain TeX
%% doesn't seem to have a way to do that.
Achilles runs 10$\times$ as fast as the tortoise, and the tortoise
starts 10 meters ahead. Where does he catch up?
$$
x_{catchup} = \sum_{i = -1}^{\infty} {10^{-i}}m = \left(10 + 1 + {1 \over 10} + {1 \over 100} + \ldots\right)m
$$
Summing infinite series is a little tricky, I forget how to do it. Maybe this will work?
$$
\eqalign {
10 x_{catchup} = 10 & \sum_{i = -1}^{\infty} {10^{-i}} m \cr
= & \sum_{i = -1}^{\infty} 10 \cdot {10^{-i}} m \cr
= & \sum_{i = -1}^{\infty} {10^{1-i}} m \cr
= & \sum_{i = -2}^{\infty} {10^{-i}} m \cr
= 10^2 + & \sum_{i = -1}^{\infty} {10^{-i}} m
}
$$
$$
\eqalign {
10 x_{catchup} - x_{catchup} & = 10^2 m + \sum_{i = -1}^{\infty} {10^{-i}} m - \sum_{i = -1}^{\infty} {10^{-i}} m \cr
& = 10^2 \cr
9 x_{catchup} & = 10^2 m \cr
x_{catchup} & = {10^2 m \over 9} \cr
& = 11{1 \over 9} m
}
$$
\bye