A Taylor series is an infinite sum of terms that represents a function. It is used to approximate a function by adding up terms that represent the value of the function and its derivatives at a specific point.
Taylor series are used in many different fields of science and engineering, such as physics, chemistry, and economics. They are useful for approximating complicated functions and solving differential equations.
The formula for a Taylor series is f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ... where f(x) is the function, a is the point of approximation, and f'(a), f''(a), f'''(a), etc. are the derivatives of the function evaluated at a.
A Taylor series is a generalization of the Maclaurin series, which is a Taylor series centered at a = 0. In other words, a Maclaurin series is a special case of a Taylor series where the point of approximation is 0.
The accuracy of a Taylor series approximation depends on the number of terms used in the series. The more terms used, the more accurate the approximation will be. Additionally, the error of a Taylor series approximation can be estimated using the remainder term formula, which takes into account the value of the next term in the series.