# What is the radius of convergence?

Given a real power series ${\sum}_{n = 0}^{+ \infty} {a}_{n} {\left(x - {x}_{0}\right)}^{n}$, the radius of convergence is the quantity r = "sup" \{tilde{r} \in \mathbb{R} : sum_{n=0}^{+infty}a_n tilde{r}^n " converges"\}. Note that $r \ge 0$, because for $t i l \mathrm{de} \left\{r\right\} = 0$ the series ${\sum}_{n = 0}^{+ \infty} {a}_{n} t i l \mathrm{de} {\left\{r\right\}}^{n} = {\sum}_{n = 0}^{+ \infty} {a}_{n} {0}^{n} = 1$ converges (recall that ${0}^{0} = 1$).

This quantity it's a bound to the value taken by $| x - {x}_{0} |$. It's not hard to prove that the given power series will converge for every $x$ such that $| x - {x}_{0} | < r$ and it will not converge if $| x - {x}_{0} | > r$ (the proof is based on the direct comparison test). The convergence of the case $r = | x - {x}_{0} |$ depends on the specific power series.

This means that the interval $\left({x}_{0} - r , {x}_{0} + r\right)$ (the interval of convergence) is the interval of the values of $x$ for which the series converges, and there are no other values of $x$ for which this happens, except for the two endpoints ${x}_{+} = {x}_{0} + r$ and ${x}_{-} = {x}_{0} - r$ for which the convergence has to be tested case-by-case.
The term radius is thereby appropriate, because $r$ describes the radius of an interval centered in ${x}_{0}$.

The definition of radius of convergence can also be extended to complex power series.