Taylor’s Formula

Taylor’s Formula is a mathematical expression used to approximate a function by an infinite sum of its derivatives evaluated at a single point. This formula serves as the foundation for the Taylor series and the Maclaurin series, which are widely used in calculus and mathematical analysis.

Taylor’s Formula states that for a function f that is n times differentiable at a point c, the function can be approximated by a polynomial of degree n centered around c. Specifically, f(x) ≈ f(c) + f'(c)(x – c) + (f”(c)/2!)(x – c)² + … + (f^n(c)/n!)(x – c)^n. The error in this approximation is given by the remainder term, which depends on the (n+1)th derivative of the function and the difference between x and c.

Taylor’s Formula is a powerful tool for approximating functions and solving differential equations. When the point c is equal to zero, the resulting series is called a Maclaurin series. Taylor’s Formula has many applications in mathematics, physics, engineering, and other fields, where it can be used to study the behavior of functions and their derivatives.