# Properties of Power Series

- Author:
- Zachry Engel

## 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.
and the radius of convergence is

**1)**The series converges for all x, in which the interval of converges 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.
converges to on I
for all terms of the power series . This series converges to for all in I. When x=0, the series converges to
, 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.

**1. Sum and Difference**: The power series**2. Multiplication by a Power:**Suppose m is an integer such that**3. Composition:**If## Differentiating and Integrating Power Series

Suppose the power series converges for |x-a|<R and defines a function f on that interval
for |x-a|<R
for |x-a|<R, where C is an arbitrary constant.

**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**2.**The indefinite integral of f is found by integrating the power series for f term by term; that is,