POWER SERIES

in mathematics, infinite series that can be thought of as a polynomial with an infinite number of terms, such as 1 + x + x2 + x3 + . . . ad infinitum. Usually, a given power series will converge (that is, approach a finite sum) for all values of x less than a certain constant and diverge for all values greater than that constant (See also convergence). This constant can be determined by the ratio test for infinite series (q.v.): If a0 + a1x + a2x2 + . . . represents a general power series with given coefficients ai, then, by the ratio test, the series will converge for all values of x such that called the radius of convergence. For instance, the series 1 + x + x2 + x3 + . . . has a radius of convergence of 1 and is called the geometric series, being equal to 1/(1 - x) in closed form. The series 1 + x/1! + x2/2! + x3/3! + . . . converges for all x the magnitude of which is less than so that the series converges for any value of x. Most functions can be represented by a power series in some interval. The coefficients of such a series can be determined by the method of undetermined coefficients thus: If f(x) = a0 + a1x + a2x2 + . . . , then it follows that f(0) = a0, f(0) = a1, f(0) = 2a2, and, in general, the ith derivative satisfies f(i)(0) = i!ai. For example, if f(x) = sin x, then f(0) = sin 0 = 0, f(0) = cos 0 = 1, f(0) = -sin 0 = 0, f(0) = -cos 0 = -1, etc., giving the series for sin x as x - x3/3! + x5/5! - . . . , which converges for any value of x. Although a series may converge for all values of x, the convergence may be so slow for some values that using it to approximate a function will require a large number of terms. Instead of powers of x, powers of (x - c) can be used, in which c is some value near the desired value to be computed, such that the derivatives of the function can be calculated for the value c. In this case, the coefficients of the power series will be ai = f(i)(c)/i!. To use this for calculating sin 65, for example, let c = p/3 radians (=60); then x - c = p/36 radians (= 5), giving sin 65 = 3/2 + (1/2)(p/36) (3/2)(p/36)2 - . . . . Power series are useful for approximating functions as above, for calculating constants such as p and e, and for solving differential equations by the method of undetermined coefficients.