Taylor Series

A special case of a power series where the coefficients are specifically derived from the function’s derivatives at a particular point.

A representation of a function as an infinite sum of terms that are calculated from the values of the function’s derivatives at a single point.

The Taylor series of a function \(f(x)\) about a point \(x = a\) is given by:

\[f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + ... = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n \]

Where \(f^{(n)}(a)\) is the \(n\)-th derivative of \(f(x)\) evaluated at \(x = a\). The higher the order of the derivative, the more accurate the approximation of the function around the point \(x = a\).

Maclaurin series when the point of consideration is \(x = 0\).