Summary of Power Series and Functions

For a power series centered at $x = a$ , 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 [latex]|x-a|R[/latex]. 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 $\displaystyle\sum _{n=0}^{\infty }{x}^{n}=\frac{1}{1-x}$ for $|x|<1$ allows us to represent certain functions using geometric series.

Key Equations

Power series centered at $x=0$
$\displaystyle\sum _{n=0}^{\infty }{c}_{n}{x}^{n}={c}_{0}+{c}_{1}x+{c}_{2}{x}^{2}+\cdots$
Power series centered at $x=a$
$\displaystyle\sum _{n=0}^{\infty }{c}_{n}{\left(x-a\right)}^{n}={c}_{0}+{c}_{1}\left(x-a\right)+{c}_{2}{\left(x-a\right)}^{2}+\cdots$

interval of convergence: the set of real numbers x for which a power series converges

power series: a series of the form $\displaystyle\sum _{n=0}^{\infty }{c}_{n}{x}^{n}$ is a power series centered at $x=0$ ; a series of the form $\displaystyle\sum _{n=0}^{\infty }{c}_{n}{\left(x-a\right)}^{n}$ is a power series centered at $x=a$

radius of convergence: if there exists a real number $R>0$ such that a power series centered at $x=a$ converges for [latex]|x-a|R[/latex], then R is the radius of convergence; if the power series only converges at $x=a$ , the radius of convergence is $R=0$ ; if the power series converges for all real numbers x, the radius of convergence is $R=\infty$