%% Remaining problems:
%% - how do I get old-style numbers by default?
\documentclass{article}
\usepackage{amsmath}      % to get the ``align'' environment and \text
%\usepackage[osf]{mathpazo}    % I wish I could get old-style numbers by default
\usepackage[hmargin=1in,vmargin=1in]{geometry}
\title{trying to remember how to sum infinite geometric series}
\author{Kragen Javier Sitaker}
\begin{document}
   \maketitle
\newcommand{\m}{\text{m}}       % for ``meter''

Achilles runs 10$\times$ as fast as the tortoise, and the tortoise
starts $10\m$ ahead. Where is the point $x_c$ where he catches up?

%% Maybe I should be using \begin{split} here instead?
\begin{align*}                  % like \begin{align} without the numbering
x_{c} & = \sum_{i = -1}^{\infty} {10^{-i}}\m \\
     & = \left(10 + 1 + \frac{1}{10} + \frac{1}{100} + \dotsb\right)\m
\end{align*}

Summing infinite series is a little tricky, I forget how to do it. Maybe this will work?

\begin{align*}
10 x_{c} = 10 & \sum_{i = -1}^{\infty} {10^{-i}} \m \\
        = & \sum_{i = -1}^{\infty} 10 \cdot {10^{-i}} \m \\
        = & \sum_{i = -1}^{\infty} {10^{1-i}} \m \\
        = & \sum_{i = -2}^{\infty} {10^{-i}} \m \\
        = 10^2 + & \sum_{i = -1}^{\infty} {10^{-i}} \m
\end{align*}

\begin{align*}
10 x_{c} - x_{c} & = 10^2 \m + \sum_{i = -1}^{\infty} {10^{-i}} \m - \sum_{i = -1}^{\infty} {10^{-i}} \m \\
                & = 10^2 \\
         9 x_{c} & = 10^2 \m \\
           x_{c} & = \frac{10^2 \m}{9} \\
                & = 11\frac{1}{9} \m
\end{align*}

Yup.

\LaTeX\ was a little better than Plain \TeX\ for this.

\end{document}