The range of convergence refers to the set of values within which a mathematical series or sequence converges. In the context of power series, it represents a specific interval or region of the real number line where the series converges and has a finite value.
Mathematically, the range of convergence of a power series is determined by the behavior of its terms as the variable approaches different values. Typically, this analysis involves examining the ratio between consecutive terms or utilizing the root test to establish convergence properties.
If a power series converges for all values of the variable x, then its range of convergence is said to be the entire real number line. Conversely, if the series diverges for all values of x, the range of convergence is considered empty. Otherwise, the range of convergence may be a finite interval, an open interval, a half-open interval, or a single value.
The concept of the range of convergence is closely related to the concept of the radius of convergence, which denotes the distance from the center of a power series to the nearest point at which the series either diverges or fails to converge.
Understanding the range of convergence is crucial for determining the validity and application of power series in various mathematical contexts, including calculus, analysis, and differential equations. This concept enables mathematicians to establish the set of values where a power series significantly approximates the function it represents, leading to valuable insights and computations in scientific and engineering fields.
