The radius of convergence of a power series f centered on a point a is equal to the distance from a to the nearest point where f cannot be defined in a way that makes it holomorphic.

Nov 11, 2025 · In this section we will give the definition of the power series as well as the definition of the radius of convergence and interval of convergence for a power series.

Apr 24, 2024 · The radius of convergence is a concept in mathematics, particularly in the study of power series. It refers to the distance from the centre of a power series to the nearest point …

Aug 22, 2023 · To find the radius of convergence, apply the Ratio Test: Determine the limit of the absolute value of the ratio of consecutive coefficients as the term number approaches infinity. …

May 27, 2025 · The radius of convergence is the distance from the center of a power series to the nearest point where the series diverges. It defines the interval within which the series …

For complex-valued functions, the circle of convergence (also called the region of convergence) is the set of values for which a power series is valid [1]. The circle of convergence is basically the …

Consider the power series: The radius of convergence “R” is any number such that the power series will converge for |x – a| < R and diverge for |x – a| > R. The power series may not …

Feb 20, 2025 · Definition: A power series a 0 +a 1 x+a 2 x 2 +… = ∑ a n x n is said to have R as radius of convergence if the power series converges absolutely for |x|<R and diverges for …

Dec 3, 2025 · In general, there is always an interval (-R,R) in which a power series converges, and the number R is called the radius of convergence (while the interval itself is called the …

The maximum value of |x| at which the series converges is called its radius of convergence. It is obvious that the same series represents a convergent and infinitely differentiable function for …