6.1 Power Series and Functions

For a power series centered at x=a,x=a, one of the following three properties hold:
1. The power series converges only at $x = a .$ In this case, we say that the radius of convergence is $R = 0 .$
2. The power series converges for all real numbers x. In this case, we say that the radius of convergence is $R = \infty .$
3. There is a real number R such that the series converges for $| x - a | < R$ and diverges for $| x - a | > R .$ In this case, the radius of convergence is R.
If a power series converges on a finite interval, the series may or may not converge at the endpoints.
The ratio test may often be used to determine the radius of convergence.
The geometric series $\sum_{n = 0}^{\infty} x^{n} = \frac{1}{1 - x}$ for $| x | < 1$ allows us to represent certain functions using geometric series.

6.2 Properties of Power Series

Given two power series $\sum_{n = 0}^{\infty} c_{n} x^{n}$ and $\sum_{n = 0}^{\infty} d_{n} x^{n}$ that converge to functions f and g on a common interval I, the sum and difference of the two series converge to $f \pm g,$ respectively, on I. In addition, for any real number b and integer $m \geq 0,$ the series $\sum_{n = 0}^{\infty} b x^{m} c_{n} x^{n}$ converges to $b x^{m} f (x)$ and the series $\sum_{n = 0}^{\infty} c_{n} {(b x^{m})}^{n}$ converges to $f (b x^{m})$ whenever bx^m is in the interval I.
Given two power series that converge on an interval $(− R, R),$ the Cauchy product of the two power series converges on the interval $(− R, R) .$
Given a power series that converges to a function f on an interval $(− R, R),$ the series can be differentiated term-by-term and the resulting series converges to $f^{'}$ on $(− R, R) .$ The series can also be integrated term-by-term and the resulting series converges to $\int f (x) d x$ on $(− R, R) .$

Taylor polynomials are used to approximate functions near a value $x = a .$ Maclaurin polynomials are Taylor polynomials at $x = 0 .$
The nth degree Taylor polynomials for a function $f$ are the partial sums of the Taylor series for $f .$
If a function $f$ has a power series representation at $x = a,$ then it is given by its Taylor series at $x = a .$
A Taylor series for $f$ converges to $f$ if and only if $lim_{n \to \infty} R_{n} (x) = 0$ where $R_{n} (x) = f (x) - p_{n} (x) .$
The Taylor series for e^x, $sin x,$ and $cos x$ converge to the respective functions for all real x.

The binomial series is the Maclaurin series for $f (x) = {(1 + x)}^{r} .$ It converges for $| x | < 1 .$
Taylor series for functions can often be derived by algebraic operations with a known Taylor series or by differentiating or integrating a known Taylor series.
Power series can be used to solve differential equations.
Taylor series can be used to help approximate integrals that cannot be evaluated by other means.