Taylor Series
The Taylor series, or Taylor expansion of a function, is defined as
Definition: Taylor Series
$$
\begin{equation*}
\sum\limits_{k=0}^{\infty} \dfrac{ f^{(k)} \left( c \right) }{k!} \left( x - c \right)^{k}.
\end{equation*}
$$
This is a infinite series of powers of the variable $x$, which is convergent for some values of $x$ such that $\left| x - c \right| < r $ where $r$ is the radius of convergence.
Theorem: Taylor's Theorem
$$
\begin{equation*}
f\left( x \right) = \sum\limits_{k=0}^{n} \dfrac{f^{(k)} \left(c\right) }{k!} \left( x- c \right)^{k} + \dfrac{f^{(n+1)} \left( \xi \right) }{\left( n + 1 \right)!} \left( x - c \right)^{n+1}
\end{equation*}
$$$$
\begin{equation*}
\lim\limits_{\xi \rightarrow c} \dfrac{ f^{(n+1)} \left( \xi \right) }{ \left( n + 1 \right)!} \left( x - c \right)^{n+1} = 0.
\end{equation*}
$$
Theorem: Taylor's Theorem for Multivariate Functions
$$
\begin{equation*}
f\left( \boldsymbol{x} + \boldsymbol{a} \right) = f\left( \boldsymbol{x} \right) + \boldsymbol{a}\cdot J + \dfrac{1}{2} \boldsymbol{a}^T H \boldsymbol{a} + \ldots
\end{equation*}
$$
where $J$ is the Jacobian and $H$ is the Hessian.
Example:
Theorem: Rolle's Theorem
$$
f^\prime (c) = 0.
$$
Theorem: Mean Value Theorem
$$
f^\prime (c) = \dfrac{f(b) - f(a)}{b - a}.
$$