I have colored my fences \right( and \left( by grouping my fences into the same scope by following a comment here by @Bardaigorl.
Is there a better way to accomplish my goal of getting the fences the same color?
\documentclass[11pt]{amsart}
\usepackage{amsmath,amssymb}
\usepackage{graphicx}
\usepackage{tikz} % Loads the xcolor package.
\newcommand{\N}{\mathbb{N}} % The natural numbers.
\newcommand{\eu}{\ensuremath{\mathrm{e}}} % The Euler number, `e'.
\newcommand{\iu}{\ensuremath{\mathrm{i}}} % The imaginary unit `i'.
\newcommand{\du}{\ensuremath{\mathop{}\!\textnormal{\slshape d}}} % A slant roman differential operator.
%%% The following redefinition of \textcolor [from stackexchange] preserves the spacing around binary operators in mathmode.
\makeatletter
\renewcommand*{\@textcolor}[3]{%
\protect\leavevmode
\begingroup
\color#1{#2}#3%
\endgroup
}
\makeatother
\begin{document}
Let's see what we can do with Euler's formula, $\eu^{\vartheta \iu } = \cos{\vartheta} + \iu \sin{\vartheta}$ when we think in terms of a very, very long polynomial, and include appropriate substitutions for each occurrence of $\iu^{n }$ for $n \in \N$.
\[
\eu^{\vartheta \iu } = 1 \textcolor{red}{-} \frac{\vartheta^2}{2!} + \frac{\vartheta^4}{4!} \textcolor{red}{-} \frac{\vartheta^6 }{6!} + \frac{\vartheta^8}{8!} \textcolor{red}{-} \frac{\vartheta^{10}}{10!}+ \cdots + \textcolor{green!65!black}{\iu} {\color{green!65!black}\left( {\color{black}\vartheta \textcolor{red}{-} \frac{\vartheta^3}{3!} + \frac{\vartheta^5}{5!} \textcolor{red}{-} \frac{\vartheta^7}{7!} + \frac{\vartheta^9}{9!} \textcolor{red}{-} \frac{\vartheta^{11}}{11!} + \cdots} \right)}.
\]
\end{document}
