A power series is an infinite series of the form ∑n=0∞ cn(x-a)n, where cn are constants, x is a variable, and a is a fixed point. When the variable x is a vector, the series is known as a power series of a scalar function with a vector.
The purpose of using a power series is to approximate a function that may be difficult or impossible to compute directly. By using a finite number of terms in the series, we can get a good approximation of the function's values at any point within a certain radius of convergence.
The radius of convergence is determined by calculating the limit of the ratio of consecutive terms in the power series. If this limit exists, it represents the distance from the center point within which the series will converge.
No, a power series may only represent functions that are analytic, meaning they can be expressed as a power series. This includes most common functions such as polynomials, exponential, trigonometric, and logarithmic functions.
To approximate a function with a vector as input, we can evaluate the power series at the given point and use a finite number of terms to get an approximation of the function's value. By increasing the number of terms used, we can improve the accuracy of the approximation.