A power series has the general form[br][math]\sum_{k=0}^{\infty}c_k\left(x-a\right)^k[/math][br]where a and [math]c_k[/math] are real numbers and x is a variable. The [math]c_k[/math]'s are the [b]coefficients [/b]of the power series and a is the [b]center[/b] of the power series. The set of values of x for which the series converges is its [b]interval of convergence[/b]. The[b] radius of convergence[/b] of the power series, denoted [i]R[/i] is the distance from the center of the series to the boundary of the interval of convergence.

A power series [math]\sum_{k=0}^{\infty}c_k\left(x-a\right)^k[/math] centered at a converges in one of three ways.[br][b]1) [/b]The series converges for all x, in which the interval of converges is [math]\left(-\infty,\infty\right)[/math] and the radius of convergence is [math]R=\infty[/math][br][br][b]2) [/b]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.[br][br][b]3)[/b] The series converges only at a, in which case the radius of convergence is R=0

Suppose the power series [math]\sum c_kx^k[/math] and [math]\sum d_kx^k[/math] converges to f(x) and g(x) respectively, on an interval I.[br][br][b]1. Sum and Difference[/b]: The power series [math]\sum\left(c_k\pm d_k\right)x^k[/math] converges to [math]f\left(x\right)\pm g\left(x\right)[/math] on I[br][br][b]2. Multiplication by a Power: [/b]Suppose m is an integer such that [math]k+m\ge0[/math] for all terms of the power series [math]x^m\sum c_kx^k=\sum c_kx^{k+m}[/math]. This series converges to [math]x^mf\left(x\right)[/math] for all [math]x\ne0[/math] in I. When x=0, the series converges to [math]lim_{x\longrightarrow0}x^mf\left(x\right)[/math][br][br][b]3. Composition: [/b]If[math]h\left(x\right)=bx^m[/math], where m is a positive integer and b is a nonzero real number, the power series [math]\sum c_k\left(h\left(x\right)\right)^k[/math] converges to the composite function [math]f\left(h\left(x\right)\right)[/math], for all x such that h(x) is in I.

Suppose the power series [math]\sum c_k\left(x-a\right)^k[/math] converges for |x-a|<R and defines a function f on that interval[br][br][b]1.  [/b]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 [br][math]f'\left(x\right)=\sum kc_k\left(x-a\right)^{k-1}[/math][br]for |x-a|<R[br][br][b]2.[/b] The indefinite integral of f is found by integrating the power series for f term by term; that is,[br][math]\int f\left(x\right)dx=\sum c_k\frac{\left(x-a\right)^{k+1}}{k+1}+C[/math][br]for |x-a|<R, where C is an arbitrary constant.