## Summary of Power Series and Functions

### Essential Concepts

• 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 $|x-a|<R$ 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 $\displaystyle\sum _{n=0}^{\infty }{x}^{n}=\frac{1}{1-x}$ 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 $x=a$

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

## Glossary

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 $x=a$
if there exists a real number $R>0$ such that a power series centered at $x=a$ converges for $|x-a|<R$ and diverges for $|x-a|>R$, 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 $R=\infty$