What is the radius of convergence?

The radius of convergence is the distance from the center of a power series within which the series converges absolutely. For a power series Σaₙ(x-c)ⁿ centered at c, the radius R determines the interval (c-R, c+R) where the series converges.

How do you find the radius of convergence?

The radius of convergence can be found using several tests: the Ratio Test (R = lim |aₙ/aₙ₊₁|), the Root Test (R = 1/lim sup |aₙ|^(1/n)), or the Cauchy-Hadamard theorem. The ratio test is most commonly used when the limit exists.

What's the difference between radius and interval of convergence?

The radius of convergence (R) is a single number representing the distance from the center where convergence is guaranteed. The interval of convergence is the actual set of x-values where the series converges, which may include or exclude the endpoints at x = c±R.

When should I use the ratio test vs root test?

Use the ratio test when consecutive coefficients have a clear pattern and the limit lim |aₙ/aₙ₊₁| exists. Use the root test when dealing with coefficients that involve nth powers or when the ratio test is inconclusive. The root test is more general but can be harder to compute.

What happens at the endpoints of convergence?

At the endpoints x = c±R, the series may converge, diverge, or conditionally converge. You must test each endpoint separately using convergence tests like the alternating series test, p-series test, or integral test to determine the complete interval of convergence.

Can a power series have infinite radius of convergence?

Yes, some power series converge for all real numbers, giving R = ∞. Examples include the exponential series e^x = Σ(xⁿ/n!), sine series, and cosine series. This occurs when the coefficients decrease very rapidly, such as with factorial denominators.

Does the radius of convergence definitively exist for every power series?

Yes. According to the Cauchy-Hadamard theorem, every single power series has a uniquely defined radius of convergence R, where 0 ≤ R ≤ ∞. The series will absolutely converge for |x - c| R.

What happens if the ratio test limit is exactly equal to 1?

If the limit in the ratio test equals exactly 1, the test is strictly inconclusive. In the context of finding the radius of convergence, a limit involving the variable 'x' equal to 1 is exactly what defines the endpoints of your interval. You must then manually check those specific endpoint values using boundary tests like the Alternating Series Test.

Is it mathematically possible for a power series to converge only at its center point?

Yes. If the coefficients in the power series grow extremely rapidly (for example, a_n = n!), the radius of convergence will be exactly R = 0. In this degenerate scenario, the series diverges everywhere except at the single distinct point x = c (the center), where every term after the first becomes zero.

How do I find the full interval of convergence after finding the radius?

Once you calculate the radius R for a series centered at c, you know it absolutely converges on the open interval (c-R, c+R). To find the full interval, you must individually plug in x = c-R and x = c+R into the original infinite series, and run standard convergence tests (like the divergent test or alternating series test) on those numerical edges.

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Calculus Tool

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: February 2, 2026

Ratio and root test calculations

Interval of convergence determination

Step-by-step convergence analysis

Need a custom calculus calculator for your educational platform? Get a Quote

Radius of Convergence Calculator

Calculate the radius and interval of convergence for power series

Power Series Coefficients

Enter coefficients separated by commas (a₀, a₁, a₂, ...). Use decimals or fractions.

Center Point (c)

Center of the power series Σaₙ(x-c)ⁿ

Convergence Test Method

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

Ratio Test:
R = lim |aₙ / aₙ₊₁|

Root Test:
R = 1 / lim sup |aₙ|^(1/n)

Cauchy-Hadamard:
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.

📊 Convergence Diagram

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.

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Example Analysis

Exponential Series Analysis

Let's analyze the convergence of e^x = Σ(xⁿ/n!)

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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Calculus Tool

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: February 2, 2026

Ratio and root test calculations

Interval of convergence determination

Step-by-step convergence analysis

Need a custom calculus calculator for your educational platform? Get a Quote

Radius of Convergence Calculator

Calculate the radius and interval of convergence for power series

Power Series Coefficients

Enter coefficients separated by commas (a₀, a₁, a₂, ...). Use decimals or fractions.

Center Point (c)

Center of the power series Σaₙ(x-c)ⁿ

Convergence Test Method

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

The Mathematical Tests

Ratio Test:
R = lim |aₙ / aₙ₊₁|

Root Test:
R = 1 / lim sup |aₙ|^(1/n)

Cauchy-Hadamard:
R = 1 / lim sup |aₙ|^(1/n)

📊 Convergence Diagram

Shows convergence regions and interval boundaries for power series

Mathematical Foundation

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 Platform

Example Analysis

Exponential Series Analysis

Let's analyze the convergence of e^x = Σ(xⁿ/n!)

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.