A power series is an infinite series of the form ∑n=0^∞ cn(x-a)n, where cn and a are constants and x is a variable. It is used to represent a function as a polynomial, and can be used to approximate the value of the function at any point within its interval of convergence.
The interval of convergence for a power series can be found by using the ratio test, where you take the limit as n approaches infinity of the absolute value of (cn+1/cn). If this limit is less than 1, the series converges absolutely. If it is greater than 1, the series diverges. If it is equal to 1, the test is inconclusive and other methods may need to be used.
A power series is a type of Taylor series, where the coefficients cn are given by the formula cn = f(n)(a)/n!, where f(x) is the function being represented and a is the center of the series. However, Taylor series can also include terms with negative powers, while power series only have terms with non-negative powers.
No, a power series can only represent functions that are analytic within its interval of convergence. This means that the function must have a continuous derivative of all orders within that interval.
Power series can be used to approximate functions by using a finite number of terms from the series. The more terms that are used, the more accurate the approximation will be. This is particularly useful for evaluating functions that are difficult to evaluate directly, such as trigonometric or exponential functions.