For a power series centered at [latex]x=a[/latex], one of the following three properties hold:
The power series converges only at [latex]x=a[/latex]. In this case, we say that the radius of convergence is [latex]R=0[/latex].
The power series converges for all real numbers x. In this case, we say that the radius of convergence is [latex]R=\infty [/latex].
There is a real number R such that the series converges for [latex]|x-a|<R[/latex] and diverges 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 [latex]\displaystyle\sum _{n=0}^{\infty }{x}^{n}=\frac{1}{1-x}[/latex] for [latex]|x|<1[/latex] allows us to represent certain functions using geometric series.
the set of real numbers x for which a power series converges
power series
a series of the form [latex]\displaystyle\sum _{n=0}^{\infty }{c}_{n}{x}^{n}[/latex] is a power series centered at [latex]x=0[/latex]; a series of the form [latex]\displaystyle\sum _{n=0}^{\infty }{c}_{n}{\left(x-a\right)}^{n}[/latex] is a power series centered at [latex]x=a[/latex]
radius of convergence
if there exists a real number [latex]R>0[/latex] such that a power series centered at [latex]x=a[/latex] converges for [latex]|x-a|<R[/latex] and diverges for [latex]|x-a|>R[/latex], then R is the radius of convergence; if the power series only converges at [latex]x=a[/latex], the radius of convergence is [latex]R=0[/latex]; if the power series converges for all real numbers x, the radius of convergence is [latex]R=\infty [/latex]
