Properties of Power Series
Power Series
A power series has the general form
where a and are real numbers and x is a variable. The 's are the coefficients of the power series and a is the center of the power series. The set of values of x for which the series converges is its interval of convergence. The radius of convergence of the power series, denoted R is the distance from the center of the series to the boundary of the interval of convergence.
Convergence of Power Series
A power series centered at a converges in one of three ways.
1) The series converges for all x, in which the interval of converges is and the radius of convergence is
2) There is a real number R>0 such that the series converges for |x-a|<R and diverges for |x-a|>R, in which case the radius of convergence is R.
3) The series converges only at a, in which case the radius of convergence is R=0
Combining Power Series
Suppose the power series and converges to f(x) and g(x) respectively, on an interval I.
1. Sum and Difference: The power series converges to on I
2. Multiplication by a Power: Suppose m is an integer such that for all terms of the power series . This series converges to for all in I. When x=0, the series converges to
3. Composition: If, where m is a positive integer and b is a nonzero real number, the power series converges to the composite function , for all x such that h(x) is in I.
Differentiating and Integrating Power Series
Suppose the power series converges for |x-a|<R and defines a function f on that interval
1. Then f is differentiable (which implies continuous) fro |x-a|<R and f' is found by differentiating the power series for the f term; that is
for |x-a|<R
2. The indefinite integral of f is found by integrating the power series for f term by term; that is,
for |x-a|<R, where C is an arbitrary constant.