The Radius of Convergence Calculator is a powerful tool used in calculus, mathematical analysis, and differential equations to determine the interval where a power series converges. Power series play a central role in advanced mathematics, enabling functions to be expressed as infinite sums. However, these series do not always converge everywhere; they converge only within a specific distance from the center, known as the radius of convergence.

What Is a Radius of Convergence?

A radius of convergence refers to the distance from the center of a power series within which the series converges. For a power series of the form:

a₀ + a₁(x − c) + a₂(x − c)² + a₃(x − c)³ + …

the series converges when:

|x − c| < R
and diverges when:
|x − c| > R

Here, R is the radius of convergence.

This concept is essential for:

• Taylor series expansions
• Maclaurin series
• Solving differential equations
• Approximation of functions
• Complex analysis

The Radius of Convergence Calculator eliminates lengthy work by calculating R quickly and accurately.

How the Radius of Convergence Calculator Works

The tool uses standard series convergence tests to determine R, typically applying:

Ratio Test Formula (Plain Text)

R = 1 / limit as n→∞ of |aₙ₊₁ / aₙ|

Root Test Formula (Plain Text)

R = 1 / limit as n→∞ of nth root of |aₙ|

Depending on the sequence provided, the calculator determines the correct limit and solves for the radius.

How to Use the Radius of Convergence Calculator

Step 1: Enter the coefficient function

Provide the expression for aₙ, the n-th term of the series. It could be something like 1/n!, n², 3ⁿ, etc.

Step 2: Enter the variable center (optional)

If the series is centered at c instead of 0, provide its value.

Step 3: Choose the method (Ratio or Root Test)

Most users leave this on automatic unless they want a specific approach.

Step 4: Click Calculate

The tool evaluates the limit and outputs the radius of convergence.

Step 5: View the interval of convergence

Some calculators also show the interval (R interval), but always check endpoints manually.

This straightforward process makes the tool ideal for students, teachers, and professionals working with power series.

Radius of Convergence Formulas Explained

Understanding the formulas behind the calculator helps build mathematical intuition.

1. Using the Ratio Test

Plain text formula:
R = 1 / limit as n→∞ of |aₙ₊₁ / aₙ|

This test is commonly used because coefficients often simplify well when using ratios.

Example:

If aₙ = 1/n, then:
aₙ₊₁ / aₙ = (1/(n+1)) / (1/n)
= n/(n+1)

Take the limit as n→∞:
limit = 1

Thus:
R = 1 / 1 = 1

2. Using the Root Test

Plain text formula:
R = 1 / limit as n→∞ of nth root of |aₙ|

This method works best when terms involve exponentials or powers of n.

Example:

If aₙ = 3ⁿ, then:
nth root of |aₙ| = nth root of 3ⁿ = 3

So:
R = 1 / 3

3. Power Series Representation

Power series formula:
Sum from n=0 to ∞ of aₙ(x − c)ⁿ

Convergence condition:
|x − c| < R

The calculator solves for R, letting you determine where the series converges.

Examples of Radius of Convergence Calculations

Example 1: aₙ = 1 / n!

Using the ratio test:
aₙ₊₁ / aₙ = (1/(n+1)!) / (1/n!)
= 1/(n+1)

Limit = 0

So:
R = 1 / 0 = ∞

Result: The series converges everywhere.

Example 2: aₙ = n²

Ratio test:
aₙ₊₁ / aₙ = (n+1)² / n²
= (n² + 2n + 1) / n²
= 1 + 2/n + 1/n²

Limit = 1

Thus:
R = 1 / 1 = 1

Result: Converges when |x − c| < 1.

Example 3: aₙ = 5ⁿ

Root test:
nth root of aₙ = nth root of 5ⁿ = 5

R = 1 / 5

Result:
Radius of convergence = 0.2

Example 4: aₙ = 3ⁿ / n

Root test:
nth root of |3ⁿ / n| = nth root of 3ⁿ × nth root of 1/n
= 3 × nth root of (1/n)

nth root of (1/n) = 1^(1/n) / n^(1/n)

As n → ∞, n^(1/n) → 1

So:
Limit = 3

Thus R = 1/3

Helpful Information About Radius of Convergence

1. Radius ≠ Interval of convergence

R gives the distance; endpoints must be checked separately.

2. Infinite radius means the series converges everywhere

This happens with power series involving factorial terms.

3. Root Test works better with exponential expressions

Useful for terms like 2ⁿ, 3ⁿ, 5ⁿ/n, etc.

4. Ratio Test works well with factorials

Great for terms like n!, (n+1)!, or combinations.

5. Divergence outside the radius is guaranteed

If |x − c| > R, the series always diverges.

6. The center does not affect the radius

But it shifts the interval.

7. Complex analysis uses the same formula

Radius of convergence applies in complex domains.

8. Power series behave like polynomials within the radius

Smooth, infinitely differentiable, easy to work with.

20 Frequently Asked Questions (FAQs)

1. What is the radius of convergence?

It is the distance from the series center within which a power series converges.

2. What does the calculator do?

It computes the radius of convergence using the Ratio or Root Test.

3. Which formula is most common?

The Ratio Test formula R = 1 / limit of |aₙ₊₁ / aₙ|.

4. Can the radius be infinite?

Yes—common for series with factorial growth.

5. What is the interval of convergence?

The set of x-values where the power series converges.

6. Does the calculator check endpoints?

Some versions may not; endpoints must be tested separately.

7. What happens if the limit is zero?

R = ∞, meaning the series converges everywhere.

8. What happens if the limit is infinite?

R = 0, meaning the series converges only at the center.

9. Can I enter any expression for aₙ?

Yes, as long as it’s a valid sequence.

10. Does the center affect the radius?

No; it only shifts the interval location.

11. What if the sequence has no limit?

The series may diverge or require further analysis.

12. Can the Root Test be used for all series?

Yes, but it is especially powerful for exponential sequences.

13. Is the radius always positive?

Yes, except in trivial cases where R = 0.

14. Can the series converge outside the radius?

No—except possibly at endpoints.

15. Is the radius used in Taylor series?

Yes, it determines where the Taylor expansion is valid.

16. Can I find R without limits?

No, radius requires limit evaluation.

17. Is R related to absolute convergence?

Yes, radius of convergence deals with absolute convergence.

18. Does a power series behave like a polynomial inside R?

Yes, it behaves smoothly and predictably.

19. Can complex numbers be used?

Yes—radius of convergence applies in the complex plane.

20. Is this calculator useful for students?

Absolutely, especially for calculus and differential equations.