A Taylor series is a representation of a function as an infinite sum of terms, where each term is generated from the derivatives of the function evaluated at a specific point.
The Taylor series for (x-1)/(1+x) at x=1 is: (x-1)/(1+x) = -1 + 2(x-1) - 3(x-1)^2 + 4(x-1)^3 - 5(x-1)^4 + ...
A Taylor series is calculated by taking the derivatives of a function at a specific point, evaluating them at that point, and then plugging those values into the formula for a Taylor series.
The Taylor series for (x-1)/(1+x) at x=1 is useful because it allows us to approximate the value of the function at any point near x=1 by using only a finite number of terms in the series. This can be especially helpful when the function is difficult to evaluate directly.
Taylor series are used in various fields of science and engineering, such as physics, chemistry, and economics. They are used to approximate complex functions and make predictions, such as in weather forecasting and financial modeling. They are also used in computer graphics and animation to create realistic curves and surfaces.