Interval of Convergence Calculator
Power Series Configuration
Series: Σ 1/n · (x - 0)ⁿ
Understanding Interval of Convergence
The interval of convergence is the set of all x values for which a power series converges. It's determined by the radius of convergence Rand requires checking the endpoints separately.
Power Series Form
Σ aₙ(x - c)ⁿ
where c is the center and aₙ are the coefficients
Finding the Radius (Ratio Test)
The ratio test gives us: R = 1/L where
L = lim(n→∞) |aₙ₊₁/aₙ|
- If L = 0: R = ∞ (converges everywhere)
- If L = ∞: R = 0 (converges only at center)
- If 0 < L < ∞: R = 1/L
Checking Endpoints
The ratio test is inconclusive at x = c ± R. We must check these separately using:
Common Convergence Tests
- • Alternating Series Test
- • p-Series Test
- • Comparison Test
- • Limit Comparison Test
Interval Notation
- • [ ] includes endpoint (converges)
- • ( ) excludes endpoint (diverges)
- • Mix based on each endpoint
Common Power Series
eˣ = Σ xⁿ/n!
R = ∞, converges for all x
1/(1-x) = Σ xⁿ
R = 1, interval: (-1, 1)
ln(1+x) = Σ (-1)ⁿ⁺¹xⁿ/n
R = 1, interval: (-1, 1]
sin(x) = Σ (-1)ⁿx²ⁿ⁺¹/(2n+1)!
R = ∞, converges for all x
About Interval of Convergence Calculator - Power Series
Find the interval and radius of convergence for power series. Apply ratio test and check endpoints for complete convergence analysis.
Our **Interval of Convergence Calculator** determines where power series converge. Enter your series coefficient pattern to find the radius R and interval of convergence. Endpoint behavior is analyzed separately. For line integrals, see our Line Integral Calculator.
Power series Σaₙ(x-c)ⁿ converge inside a radius R around center c. The ratio test gives R = 1/L where L = lim|aₙ₊₁/aₙ|. Our calculator computes R and checks endpoint convergence using appropriate series tests.
Master power series convergence—essential for Taylor series, differential equations, and complex analysis. Understand when series converge absolutely, conditionally, or diverge.
Key Features
Why Use This Tool?
Common Use Cases
Calculus: Find where Taylor series converge.
Differential Equations: Series solutions to DEs.
Complex Analysis: Analytic function properties.
Numerical Methods: Convergence of approximations.
How to Use
Select coefficient pattern aₙ
Enter center value c
Click Calculate
View radius and interval with endpoint analysis