Radius of Convergence Calculator
Calculate the radius and interval of convergence for power series using the ratio test, root test, and Cauchy-Hadamard theorem. Analyze series convergence with step-by-step mathematical solutions.
Last updated: December 15, 2024
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Enter coefficients separated by commas (a₀, a₁, a₂, ...). Use decimals or fractions.
Center of the power series Σaₙ(x-c)ⁿ
Convergence Analysis
Radius of Convergence:
R = 1.0000
Interval of Convergence:
(-1.0000, 1.0000)
Method Used:
ratio Test
Analysis:
Using the ratio test, the radius of convergence is 1.0000. The series converges absolutely within this radius from the center point.
Calculation Steps:
- Using Ratio Test: R = lim |aₙ / aₙ₊₁|
- Coefficients: [1, 1, 1, 1, 1]
- Average ratio: 1.0000
- Radius of convergence: R = 1.0000
Convergence Tests:
- • Ratio Test: R = lim |aₙ/aₙ₊₁| (when limit exists)
- • Root Test: R = 1/lim sup |aₙ|^(1/n)
- • Series converges absolutely for |x-c| < R
- • Check endpoints separately for conditional convergence
Quick Example Result
For geometric series with coefficients [1, 1, 1, 1, 1]:
R = 1, Interval: (-1, 1)
Using ratio test: lim |aₙ/aₙ₊₁| = 1
How This Calculator Works
Our radius of convergence calculator analyzes power series using established convergence tests from advanced calculus. The calculator applies the ratio test, root test, and Cauchy-Hadamard theorem to determine where infinite series converge absolutely.
The Mathematical Tests
R = lim |aₙ / aₙ₊₁|R = 1 / lim sup |aₙ|^(1/n)R = 1 / lim sup |aₙ|^(1/n)These tests determine the radius R within which the power series Σaₙ(x-c)ⁿ converges absolutely. The series converges for |x-c| < R, diverges for |x-c| > R, and requires separate testing at endpoints.
Shows convergence regions and interval boundaries for power series
Mathematical Foundation
Power series convergence is fundamental to advanced calculus and mathematical analysis. The radius of convergence determines the largest interval around the center point where the infinite series represents a well-defined function. This concept is essential for Taylor series, Fourier analysis, and complex function theory.
- Ratio test works best when consecutive coefficients have clear relationships
- Root test is more general and works when ratio test fails or is inconclusive
- Cauchy-Hadamard theorem provides the most general framework for convergence
- Endpoint behavior must be analyzed separately using other convergence tests
Sources & References
- Real Analysis: Modern Techniques and Applications - Gerald B. Folland (2nd Edition)Standard reference for series convergence theory
- Mathematical Association of America - Advanced Calculus Teaching ResourcesEducational standards for power series and convergence
- MIT OpenCourseWare - Real Analysis and Power SeriesAcademic materials on convergence tests and applications
Need help with other series calculations? Check out our power series calculator and Fourier series calculator.
Get Custom Calculator for Your PlatformExample Analysis
Given Series:
- Series: Σ(xⁿ/n!)
- Coefficients: aₙ = 1/n!
- Center: c = 0
Ratio Test Steps:
- Calculate: |aₙ/aₙ₊₁| = |(1/n!)/(1/(n+1)!)| = n+1
- Find limit: lim(n→∞) (n+1) = ∞
- Therefore: R = ∞
- Interval: (-∞, ∞)
Result: Infinite radius of convergence (R = ∞)
The exponential series converges for all real numbers, making it an entire function. This is due to the factorial in the denominator causing rapid coefficient decay.
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