Calculators6 min readUpdated: 2026-09-14
What Is the Radius of Convergence? Definition, Formulas & Properties
Definition
The radius of convergence (R) is a non-negative real number or infinity that defines the distance from the center c of a power series to the boundary of its convergence domain.
Detailed Technical Breakdown
The radius of convergence R characterizes the size of the disk or interval in which a power series converges absolutely. For any x satisfying |x - c| < R, the series converges absolutely. For any x satisfying |x - c| > R, the series diverges. The value of R is typically computed using the Ratio Test limit R = 1 / lim |aₙ₊₁ / aₙ| or the Cauchy-Hadamard root test formula R = 1 / limsup |aₙ|^(1/n).
Key Technical Specifications
- Derived via Ratio Test: R = lim_{n→∞} |aₙ / aₙ₊₁|.
- Cauchy-Hadamard formula: 1/R = limsup_{n→∞} (|aₙ|)^(1/n).
- If terms include factorials (n!), R is generally ∞ (e.g., eˣ, sin x, cos x).
- R is strictly non-negative: R ∈ [0, ∞].
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