Radius of Convergence Calculator
Calculate the radius of convergence for power series using advanced mathematical tests. Our radius of convergence calculator provides step-by-step solutions with detailed analysis.
For series ∑aₙ(x-c)ⁿ, enter the coefficients aₙ in order
Ready to Calculate
Enter your power series coefficients to determine the radius of convergence
Understanding Radius of Convergence
The radius of convergence calculator is an essential tool for analyzing power series convergence. The radius of convergence represents the distance from the center of a power series within which the series converges absolutely.
Power Series Definition
A power series is an infinite series of the form ∑(n=0 to ∞) aₙ(x-c)ⁿ, where aₙ are coefficients, c is the center, and x is the variable. Our radius of convergence calculator analyzes these series to determine convergence behavior.
Convergence Properties
- Absolute convergence within the radius of convergence
- Divergence outside the radius of convergence
- Uncertain behavior at the boundary points
- Uniform convergence on compact subsets within the radius
Convergence Tests in Our Calculator
Ratio Test (D'Alembert)
The radius of convergence calculator uses the ratio test: R = lim |aₙ/aₙ₊₁| as n approaches infinity.
- • Most commonly used test
- • Effective for factorial and exponential terms
- • Provides direct radius calculation
Root Test (Cauchy)
Our calculator applies the root test: R = 1/lim sup |aₙ|^(1/n) as n approaches infinity.
- • Effective for exponential coefficients
- • Based on Cauchy-Hadamard theorem
- • Always provides a result when applicable
Applications of Radius of Convergence Calculator
Taylor Series Analysis
Determine convergence intervals for Taylor and Maclaurin series expansions of functions.
Complex Analysis
Analyze convergence in the complex plane and determine domains of analyticity.
Numerical Methods
Establish convergence criteria for iterative algorithms and approximation methods.
How to Use the Radius of Convergence Calculator
Input Series Coefficients
Enter the coefficients of your power series in the calculator interface.
Select Test Method
Choose between ratio test, root test, or let the calculator determine the best method.
Analyze Results
Review the calculated radius and interval of convergence with detailed explanations.
Common Series Examples
Geometric Series
∑xⁿ has radius R = 1
Exponential Series
∑xⁿ/n! has radius R = ∞
Logarithmic Series
∑xⁿ/n has radius R = 1
Advanced Concepts in Radius of Convergence
Cauchy-Hadamard Theorem
The Cauchy-Hadamard theorem provides the fundamental relationship between coefficients and radius of convergence. Our radius of convergence calculator implements this theorem to ensure accurate results for all power series types.
Interval vs Radius of Convergence
While the radius determines the distance from the center where convergence occurs, the interval of convergence includes analysis of boundary points. The radius of convergence calculator helps distinguish between these concepts.
Complex Plane Analysis
In complex analysis, the radius of convergence defines a disk in the complex plane where the power series represents an analytic function. This concept is crucial for understanding function behavior and singularities.
Frequently Asked Questions
What is the difference between radius and interval of convergence?
The radius of convergence is a number that determines the distance from the center where convergence occurs, while the interval of convergence includes the specific endpoints and their convergence behavior.
When should I use the ratio test vs root test?
Use the ratio test when coefficients involve factorials or simple ratios. Use the root test when coefficients have exponential forms or when the ratio test is inconclusive.
Can a power series have infinite radius of convergence?
Yes, some power series like the exponential series ∑xⁿ/n! converge for all complex numbers, giving them an infinite radius of convergence.
How accurate is this radius of convergence calculator?
Our calculator uses precise mathematical algorithms based on established convergence tests, providing accurate results for standard power series forms with proper coefficient inputs.