Taylor Series

A Taylor series is a power series constructed to reproduce the local behavior of a smooth function about a chosen point, called the center. Its purpose is to encode the function’s value and successive derivatives into a structured expansion that captures local variation. The central task in its derivation is to find coefficients so that the series and the function share identical values and derivatives at the center.

Consider a power series centered at x = a:

\begin{equation*}\Sigma_{n=0}^{\infty} c_n (x-a)^n\end{equation*}

Evaluating this expression at x = a eliminates all terms containing x - a, leaving only the constant term. This gives c₀ = f(a). ensuring that the series matches the function’s value at the center. This step establishes the foundation for aligning higher-order behavior.

Differentiating the series once gives

\begin{equation*}\Sigma_{n=1}^{\infty} n c_n (x-a)^{n-1}\end{equation*}

Evaluating again at x = a eliminates all remaining terms except the one associated with c₁, giving c₁ = f'(a). The same reasoning applies to higher derivatives. With each differentiation, powers decrease, and numerical factors accumulate. After n differentiations, evaluating at x = a isolates a single term, producing the relation

\begin{equation*}f^{(n)}(a)=n!c_n\end{equation*}

Solving for the coefficient gives

\begin{equation*}c_n=\jfrac{f^{(n)}(a)}{n!}\end{equation*}

Substituting these coefficients into the original power series produces the Taylor series:

\begin{equation*}f(x)=\Sigma_{n=0}^{\infty}\jfrac{f^{(n)}(a)}{n!}(x-a)^n\end{equation*}

Each term in the series corresponds to a specific derivative evaluated at the center, linking the polynomial structure directly to the function’s local geometric properties, including value, slope, curvature, and higher-order variations.