A real power series about a centre, its interval of convergence, and its radius in $[0,+\infty]$

Let $(a_n)_{n\in\mathbb N}$ be a sequence of reals and let $c\in\mathbb R$. The real power series about the centre $c$ with coefficients $(a_n)$ is the series

$\sum_{n=0}^{\infty}a_n(x-c)^n$

at a real argument $x$, where powers are those of Integer powers $a^m$ and convergence is that of Series, partial sums, convergence and the sum, divergence, and the tail series. Its value, when the series converges, is called its sum at $x$. At $x=c$ the series always converges to $a_0$: the term with $n=0$ is $a_0$ because $0^0=1$, and every later term is $0$.

For $r\ge0$ let $P(r)$ mean that the series converges absolutely at every real $x$ with $|x-c|<r$. The set of such $r$ contains $0$, since the condition $|x-c|<0$ has no solutions. The radius of convergence is

$R:=\sup_{\overline{\mathbb R}}\{r\in\mathbb R:r\ge0\text{ and }P(r)\}\in[0,+\infty],$

where the supremum is taken in the extended real line of The extended real line $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$, its order, and the arithmetic that is left undefined. Thus $R$ may be a nonnegative real or $+\infty$, but never $-\infty$.

The open interval determined by the radius is

$I_R:=\{x\in\mathbb R:|x-c|<R\}.$

When $0<R<+\infty$ this is $(c-R,c+R)$, when $R=+\infty$ it is all of $\mathbb R$, and when $R=0$ it is empty. The centre still carries the convergent value $a_0$ in the last case. No endpoint is included in $I_R$; convergence at $c-R$ or $c+R$, when these are real, is a separate question.

The radius is extended-valued, but no undefined arithmetic in $\overline{\mathbb R}$ is used. Expressions such as $c\pm R$ are written only when $R$ is finite. The reciprocal conventions used in Cauchy-Hadamard are stated explicitly in Cauchy–Hadamard: the reciprocal radius is $\limsup_{k\to\infty}|a_{k+1}|^{1/(k+1)}$, with the zero and infinite cases included.