A Taylor Series Expansion is a way of approximating a function with a polynomial by using its derivatives at a specific point. It is used to calculate the values of a function at any point within its interval of convergence.
The Taylor Series Expansion is calculated by taking the derivatives of the function at a specific point and using those derivatives to create a polynomial. The more terms included in the polynomial, the more accurate the approximation will be.
The Radius of Convergence is the distance from the center of a Taylor Series Expansion to the closest point where the series still accurately approximates the function. It is represented by the variable R and determines the interval of convergence for the series.
The Radius of Convergence is determined by using the Ratio Test, which compares the absolute value of the ratio of consecutive terms in the series to the limit as n approaches infinity. If the limit is less than 1, the series converges and the value of R can be found by taking the reciprocal of the limit.
The Radius of Convergence for this function is infinite since it is a polynomial. This means that the Taylor Series Expansion for this function will converge for all values of x.