A Taylor series is a representation of a function as an infinite sum of its derivatives evaluated at a single point. Taylor series are important in calculus because they allow for the approximation of a function by an infinite sum of its derivatives evaluated at a single point. They can be used to represent a wide variety of functions, and can be used to solve differential equations and other problems in physics and engineering. Taylor series can be derived using a variety of methods, such as the power series method, the Taylor’s formula, and the Maclaurin series.
They can be used to approximate the value of a function at a given point, or to approximate the behavior of a function over a given interval. The accuracy of a Taylor series depends on the order of the series and the point at which it is evaluated. Higher order series and points closer to the function will generally yield more accurate approximations.