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

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

${\sum }_{n=0}^{\mathrm{\infty }}{a}_n(x-c{)}^n$

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

$R:={\sup }_{\bar{\mathbb{R}}}\{r\in \mathbb{R}:r\ge 0\text{ and }P(r)\}\in [0,+\mathrm{\infty }],$

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

The open interval determined by the radius is

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

When $0<R<+\mathrm{\infty }$ this is $(c-R,c+R)$, when $R=+\mathrm{\infty }$ 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 $\bar{\mathbb{R}}$ 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 ${\mathrm{lim\hspace{0.17em}sup}}_{k\to \mathrm{\infty }}\mathrm{\mid }{a}_{k+1}{\mathrm{\mid }}^{1\mathrm{/}(k+1)}$, with the zero and infinite cases included.