Calculators6 min readUpdated: 2026-09-14
What Is the Interval of Convergence in Calculus? Definition & Examples
Definition
The interval of convergence is the complete set of all real numbers x for which an infinite power series ∑ aₙ(x - c)ⁿ converges to a finite real value.
Detailed Technical Breakdown
For any power series centered at c, the set of convergent values forms an interval centered at c. If the series has a finite positive radius of convergence R > 0, the open interval is (c - R, c + R). Because the Ratio Test is inconclusive at endpoints where L = 1, each boundary point (c - R and c + R) must be evaluated separately to determine whether the final interval includes the endpoints (using square brackets [ ]) or excludes them (using parentheses ( )).
Key Technical Specifications
- Preliminary open interval: (c - R, c + R) where R is the radius of convergence.
- Four possible interval configurations: (c-R, c+R), [c-R, c+R), (c-R, c+R], or [c-R, c+R].
- If R = 0, the interval degenerates to the single point {c}.
- If R = ∞, the interval of convergence is the entire real line (-∞, ∞).
Related Tools on ToolQix
Apply this concept directly using our free client-side browser utilities: