Radius and Interval of Convergence¶

Definition¶

For a given power series $\displaystyle \sum_{n=0}^\infty c_n (x-a)^n\text{,}$ its radius of convergence $R$ is a nonnegative real number or $+\infty$ such that the series converges if $|x-a|<R$ and diverges if $|x-a|>R\text{.}$ The interval of convergence is the set of all values of $x$ for which the series converges.

Interpretation¶

The radius of convergence tells how far from the centre $a$ a power series converges. Apart from the boundary, only the distance $|x-a|$ matters: the series converges at every $x$ within distance $R$ of the centre and diverges at every $x$ further away.

The figure shows the open interval $(a-R,a+R)$ where the series converges (absolutely), the radius $R$ as an arrow from the centre, and divergence outside. The endpoints $a\pm R$ are drawn as open circles because the radius says nothing about them. The interval of convergence is $(a-R,a+R)$ together with those endpoints at which the series converges. For $a=0$ and $R=1$ all four cases occur: $\displaystyle \sum_n x^n$ converges on $(-1,1)$, $\displaystyle \sum_{n\ge1}x^n/n$ on $[-1,1)$, $\displaystyle \sum_{n\ge1}(-x)^n/n$ on $(-1,1]$, and $\displaystyle \sum_{n\ge1}x^n/n^2$ on $[-1,1]$.

Every power series has such an $R\in[0,+\infty]$$[0,+\infty ]$. The extreme cases are $R=0$$0$, where the series converges only at its centre, and $R=+\infty$$+\infty$, where it converges for every real $x$. In practice $R$ is computed with the ratio test, $R=\lim_{n\to\infty}|c_n/c_{n+1}|$ when this limit exists, or with the root test, $R=1/\limsup|c_n|^{1/n}$$1/\mathrm{limsup}|c_n{|}^{1/n}$, which always applies.

Inside the radius, the series converges uniformly on every closed interval $[a-r,a+r]$ with $r<R$$R$, and its sum can be differentiated term by term, as shown in convergence of power series.

Two slips are common. The interval is centred at $a$, not at $0$: for $\displaystyle \sum_n n(-2)^n(x-3)^n$ in the Examples, $R=\tfrac{1}{2}$ and the interval is $(\tfrac{5}{2},\tfrac{7}{2})$. And the endpoints must be tested separately, with tests for numerical series; their behaviour cannot be read from $R$.

Key Results¶

• The convergence of $\displaystyle \sum_{n=0}^\infty c_n (x-a)^n$ at $x=a-R$ and $x=a+R$ is to be determined case-by-case.

Examples¶

• The series $\displaystyle \sum_{n=0}^\infty n(-2)^n (x-3)^n$ with $R = 1/2$ centred at $a = 3$$3$, converges pointwise on $\left(5/2, 7/2\right)$ and uniformly on any closed interval within it. Its convergence interval is $\left(5/2, 7/2\right)$ since the series diverges at both endpoints.
• The series $\displaystyle \sum_{n=0}^\infty 2^{n^2} (x+2)^n$ with $R = 0$ centred at $a = -2$$-2$, converges solely at its centre $x = -2.$

Prerequisites¶

• Power Series

Related Concepts¶

• Ratio Test
• Root Test

Sources¶

• Math III: Analysis · Lecture 2

Aliases: radius of convergence, interval of convergence, convergence radius Symbols: element of, infinity, sum

Last updated: 20 October 2026 03:02