(See earlier edits if needed, but this is my cleanest version yet. References: Polyglot note-keeping?, comment package, TeX FAQ: Conditional compilation and “comments”)
This can be solved with a combination of the verbatim and extract packages. While building the PDF, the extract package will simultaneously create a synopsis.tex file with just the synopses and document divisions extracted. Build this synopsis.tex file, and you'll have your outline.
Original document source:
\documentclass{article}
% In this file, make a new comment-like environment named 'synopsis':
\usepackage{verbatim}
\let\synopsis\comment
\let\endsynopsis\endcomment
% Extract synopsis environments, \section commands, and \tableofcontents
% commands into a separate synopsis.tex file. In that file, a synopsis
% environment will be a simple semantic environment with no extra decoration.
\usepackage[active,
generate=synopsis,
extract-env={synopsis},
extract-cmd={section},
extract-cmdline={tableofcontents}]{extract}
\begin{extract}
\newenvironment{synopsis}{}{}
\end{extract}
% Here is the real document, with inline synopses:
\begin{document}
\tableofcontents
\section{Fourier Series}
\begin{synopsis}
A Fourier series is periodic, and made up of orthogonal $\sin$ and $\cos$ functions.
\end{synopsis}
The Fourier series has the form:
\[
\frac{a_0}{2} +
\sum_{n=1}^{\infty} \left[ a_n \cos {\frac{2 n \pi}{T} t} +
b_n \sin {\frac{2 n \pi}{T} t} \right]
\]
where $a_n$ and $b_n$ are called Fourier coefficients of the Fourier series
of the function $f(x)$.
\end{document}
Resulting PDF:

Generated synopsis.tex:
%%
%% This is file, `synopsis.tex',
%% generated with the extract package.
%%
%% Generated on : 2012/04/02,18:26
%% From source : 50099.tex
%% Using options: active,generate=synopsis,extract-env={synopsis},extract-cmd={section},extract-cmdline={tableofcontents}
%%
\documentclass{article}
\newenvironment{synopsis}{}{}
\begin{document}
\tableofcontents
\section{Fourier Series}
\begin{synopsis}
A Fourier series is periodic, and made up of orthogonal $\sin$ and $\cos$ functions.
\end{synopsis}
\end{document}
and PDF:
