The Taylor series is a mathematical tool that expresses a function as an infinite sum of terms calculated from its derivatives at a single point. Essentially, it approximates complicated functions using polynomials, where each term involves the function’s derivatives evaluated at the expansion point and powers of the distance from that point. For example, expanding around zero gives the Maclaurin series, a special case of the Taylor series. This method is especially useful because many functions

The Taylor series is a way of representing a function as an infinite sum of terms based on its derivatives at a single point. It allows us to approximate complicated functions using polynomials, which are easier to work with. Each term in the series involves a derivative of the function evaluated at a chosen point, usually called “a,” and is multiplied by a power of (x - a). The more terms we include, the better the approximation becomes, especially when x is close to a. This method is commonly