Convergence Radius

analysis

Definition

Convergence Radius

The convergence radius of a power series is a non-negative real number (or ) such that the series converges for all satisfying and diverges for all satisfying .

The behaviour at the boundary points must be tested separately (see Abel’s Limit Theorem).

Complex Space Representation

In the complex plane , the interval of convergence generalises to a disk of convergence. The series converges for all complex numbers lying within this disk.

Calculation

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

Definition

Cauchy-Hadamard Theorem

The convergence radius of a power series is determined by the limit superior of the absolute values of the coefficients:

where:

• If , then .
• If , then .

The theorem ensures that the series converges absolutely for all and diverges for all . It is a direct consequence of the Root Criterion.

Link to original

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

Properties

• Inside the disk of convergence (), 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.