A Taylor Series is a mathematical representation of a function using an infinite sum of terms. In the case of sin(x), the Taylor Series is a sum of terms involving the derivatives of sin(x) evaluated at a specific point, in this case π/2.
The center of the Taylor Series for sin(x) is chosen to be at π/2 because this is the value at which the derivatives of sin(x) are easiest to evaluate. At this point, sin(x) and all of its derivatives have simple and predictable values, making the series easier to work with.
The radius of convergence for the Taylor Series for sin(x) centered at π/2 is infinite. This means that the series will converge for all values of x, no matter how large or small. This is because the sine function is an entire function, meaning it is analytic on the entire complex plane.
The Taylor Series for sin(x) centered at π/2 is derived using the Taylor series formula, which involves calculating the derivatives of sin(x) at π/2 and plugging them into the formula. This results in an infinite series of terms that can be simplified and expressed in a more compact form.
The Taylor Series for sin(x) is significant because it allows us to approximate the value of sin(x) at any point using a finite number of terms. This makes it a useful tool for solving mathematical problems involving the sine function, and it also has applications in many other areas of mathematics, such as calculus and differential equations.