Convergence Radius

analysis

Definition

Convergence Radius

The convergence radius $R$ of a power series ${\sum }_{n=0}^{\infty }{a}_n(x-x_0{)}^n$ is a non-negative real number (or $\infty$) such that the series converges for all $x$ satisfying $|x-x_0|<R$ and diverges for all $x$ satisfying $|x-x_0|>R$.

The behaviour at the boundary points $x=x_0\pm R$ must be tested separately (see Abel’s Limit Theorem).

Complex Space Representation

In the complex plane $\mathbb{C}$, the interval of convergence generalises to a disk of convergence. The series converges for all complex numbers $z$ lying within this disk.

Calculation

The convergence radius can be calculated using the Cauchy-Hadamard theorem:

Definition

Cauchy-Hadamard Theorem

The convergence radius $R$ of a power series ${\sum }_{n=0}^{\infty }{a}_n(x-x_0{)}^n$ is determined by the limit superior of the absolute values of the coefficients:

$$\frac{1}{R}=\underset{n\to \infty }{\mathrm{limsup}}\sqrt[n]{|a_n|}$$

where:

• If ${\mathrm{limsup}}_{n\to \infty }\sqrt[n]{|a_n|}=0$, then $R=\infty$.
• If ${\mathrm{limsup}}_{n\to \infty }\sqrt[n]{|a_n|}=\infty$, then $R=0$.

The theorem ensures that the series converges absolutely for all $|x-x_0|<R$ and diverges for all $|x-x_0|>R$. It is a direct consequence of the Root Criterion.

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Alternatively, if the following limit exists, the Quotient Criterion can be used:

$$R=\underset{n\to \infty }{\lim }\left|\frac{a_n}{a_{n+1}}\right|$$

Properties

• Inside the disk of convergence ($|x-x_0|<R$), the power series is absolutely convergent and defines an analytic function.
• The series can be differentiated and integrated term-by-term within this interval without changing the convergence radius.