A Taylor Series is a mathematical representation of a function as an infinite sum of terms, each term being a derivative of the function evaluated at a specific point.
In the Newton-Raphson method, a Taylor Series is used to approximate the root of a function by finding the intersection point of the tangent line of the function at a given point and the x-axis.
The Newton-Raphson method is used to find the roots of a nonlinear function by iteratively improving the initial estimate of the root.
The Newton-Raphson method is generally more efficient and accurate compared to other root-finding methods for nonlinear functions. It also has a fast convergence rate, meaning it requires fewer iterations to reach a solution.
The Newton-Raphson method may fail to converge if the initial estimate is too far from the actual root or if there are multiple roots. It also requires knowledge of the derivative of the function, which may not be easily calculable for some functions.