Taylor Series

A Taylor series is a power series that represents a smooth function near a specific point called the center. It encodes the function's value and successive derivatives into a structured expansion that captures local variation. The series matches the function's value, slope, curvature, and higher-order behavior at the center point.

Taylor series coefficients are found by repeatedly differentiating the power series and evaluating at the center. Each differentiation isolates a specific coefficient and creates factorial multipliers. The nth coefficient equals the nth derivative of the function at the center divided by n factorial, giving the formula c_n = f^(n)(a) / n!.

When you evaluate a power series centered at x = a at the point x = a, all terms containing (x - a) become zero, leaving only the constant term. This reveals the first coefficient directly. Differentiating before evaluation eliminates lower-degree terms, allowing each successive derivative to isolate its corresponding coefficient.

Factorials appear as numerical multipliers when differentiating power series repeatedly. Each differentiation of (x - a)^n produces factors like n, (n-1), and so on, which multiply together to form n!. Dividing by this factorial in the coefficient formula ensures the series accurately reproduces the function's derivatives at the center.

Each term in a Taylor series corresponds to a specific derivative evaluated at the center, linking the polynomial structure directly to the function's local geometry. The constant term represents the function's value, the linear term encodes slope, the quadratic term captures curvature, and higher-order terms represent increasingly subtle variations in the function's behavior.

A Taylor series is a specific type of power series constructed to match a given function's value and derivatives at a chosen center. While power series and their properties form the general framework, Taylor series apply that framework by determining coefficients through derivatives, creating a power series that reproduces the function's local behavior.

The center point is where the Taylor series matches the function's value and all its derivatives. Choosing a specific center allows the series to accurately represent the function's behavior in a neighborhood around that point. Different centers produce different series, each optimized for approximating the function near its respective center.