Power Series - Radius of Convergence

Radius of Convergence

A power series will converge for some values of the variable x and may diverge for others. All power series f(x) in powers of (x-c) will converge at x = c. (The correct value f(c) = a₀ requires interpreting the expression 00 as equal to 1.) If c is not the only convergent point, then there is always a number r with 0 < r ≤ ∞ such that the series converges whenever |x − c| < r and diverges whenever |x − c| > r. The number r is called the radius of convergence of the power series; in general it is given as

or, equivalently,

(this is the Cauchy–Hadamard theorem; see limit superior and limit inferior for an explanation of the notation). A fast way to compute it is

if this limit exists.

The series converges absolutely for |x − c| < r and converges uniformly on every compact subset of {x : |x − c| < r}. That is, the series is absolutely and compactly convergent on the interior of the disc of convergence.

For |x − c| = r, we cannot make any general statement on whether the series converges or diverges. However, for the case of real variables, Abel's theorem states that the sum of the series is continuous at x if the series converges at x. In the case of complex variables, we can only claim continuity along the line segment starting at c and ending at x.