%% 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
