Essential Concepts
- For a power series centered at x=a, one of the following three properties hold:
- The power series converges only at x=a. In this case, we say that the radius of convergence is R=0.
- The power series converges for all real numbers x. In this case, we say that the radius of convergence is R=∞.
- 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 ∞∑n=0xn=11−x for |x|<1 allows us to represent certain functions using geometric series.
Key Equations
- Power series centered at x=0
∞∑n=0cnxn=c0+c1x+c2x2+⋯ - Power series centered at x=a
∞∑n=0cn(x−a)n=c0+c1(x−a)+c2(x−a)2+⋯
Glossary
- interval of convergence
- the set of real numbers x for which a power series converges
- power series
- a series of the form ∞∑n=0cnxn is a power series centered at x=0; a series of the form ∞∑n=0cn(x−a)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=∞
Candela Citations
CC licensed content, Shared previously
- Calculus Volume 2. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: https://openstax.org/books/calculus-volume-2/pages/1-introduction. License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at https://openstax.org/books/calculus-volume-2/pages/1-introduction