Radius of Convergence Calculator

Welcome to the Radius of Convergence Calculator, a comprehensive tool for analyzing power series convergence. Whether you are studying calculus, preparing for exams, or doing mathematical research, this calculator determines the radius and interval of convergence using either the Ratio Test or the Root Test, providing detailed step-by-step solutions with mathematical notation.

What is the Radius of Convergence?

The radius of convergence \( R \) of a power series \( \sum_{n=0}^{\infty} a_n (x - c)^n \) is the non-negative extended real number such that the series converges absolutely for \( |x - c| < R \) and diverges for \( |x - c| > R \). At the boundary \( |x - c| = R \), convergence must be checked separately at each endpoint.

The radius of convergence defines a symmetric interval around the center \( c \) within which the power series represents a well-defined function. This concept is fundamental in analysis, differential equations, and many areas of applied mathematics.

Power Series General Form

Methods for Finding the Radius of Convergence

The Ratio Test

The most commonly used method. Compute the limit:

The Ratio Test is especially effective when the general term involves factorials, exponentials, or products. It directly compares the growth rate of consecutive terms.

The Root Test (Cauchy-Hadamard Theorem)

An alternative that is sometimes more powerful:

The Root Test is particularly useful when the general term involves nth powers like \( a_n = r^n \) or expressions where the ratio of consecutive terms is hard to simplify.

How to Use This Calculator

1. Choose input mode: Enter either the general term \( a_n \) as a mathematical expression, or provide a list of coefficients.
2. Specify the center: Enter the center \( c \) of your power series (default is 0 for Maclaurin series).
3. Select the test: Choose between the Ratio Test or Root Test based on the form of your series.
4. Calculate: Click the button to see the radius of convergence, interval of convergence, step-by-step derivation, and convergence visualization.

Understanding the Results

Three Possible Outcomes

• \( R = \infty \): The series converges for all real numbers \( x \). Examples include \( e^x, \sin(x), \cos(x) \).
• \( 0 < R < \infty \): The series converges on the open interval \( (c - R, c + R) \) and diverges outside. Endpoints require separate analysis.
• \( R = 0 \): The series converges only at the center \( x = c \). Example: \( \sum n! \cdot x^n \).

Endpoint Analysis

When \( 0 < R < \infty \), the Ratio and Root Tests are inconclusive at \( x = c \pm R \). You need additional tests:

• Alternating Series Test: For series with alternating signs at endpoints
• p-Series Test: Compare with \( \sum 1/n^p \)
• Comparison Test: Compare with a known convergent or divergent series
• Divergence Test: If the terms do not approach zero, the series diverges

Common Power Series and Their Radii

When to Use Each Test

Use the Ratio Test When:

• The general term contains factorials (e.g., \( n! \), \( (2n)! \))
• The term involves products of sequential integers
• You can easily simplify the ratio \( a_{n+1}/a_n \)

Use the Root Test When:

• The general term has the form \( (f(n))^n \)
• The term involves nth powers that simplify under nth roots
• The Ratio Test is inconclusive (both tests agree when both work, but the Root Test is strictly more powerful)

Input Syntax Guide

• Powers: Use ** or ^ (e.g., n**2 or n^2)
• Factorial: Use factorial(n) (e.g., 1/factorial(n))
• Common functions: sin, cos, tan, exp, log, ln, sqrt
• Constants: pi, e
• Variable: Use n for the index variable, x for the series variable

Frequently Asked Questions

What is the radius of convergence?

The radius of convergence R of a power series is the distance from the center of the series to the boundary of the region where the series converges. For a power series centered at a, the series converges absolutely when |x - a| < R and diverges when |x - a| > R. R can be 0 (converges only at the center), a positive number, or infinity (converges everywhere).

How do you find the radius of convergence using the Ratio Test?

To find the radius of convergence using the Ratio Test: compute L = lim(n to infinity) |a_{n+1}/a_n|. The radius of convergence is R = 1/L. If L = 0, R = infinity (converges everywhere). If L = infinity, R = 0 (converges only at the center). The series converges absolutely when |x - a| < R.

What is the difference between the Ratio Test and Root Test?

Both tests determine the radius of convergence but use different approaches. The Ratio Test computes the limit of |a_{n+1}/a_n|, while the Root Test computes the limit of |a_n|^(1/n). The Root Test is sometimes more powerful (it works whenever the Ratio Test works, plus some cases where it does not), but the Ratio Test is often easier to compute for expressions involving factorials.

Does the radius of convergence tell us about the endpoints?

No. The radius of convergence only tells us about absolute convergence inside the interval and divergence outside. At the endpoints x = a - R and x = a + R, the series may converge or diverge, and each endpoint must be tested separately using other tests like the Alternating Series Test, p-series test, or Comparison Test.

What are common power series and their radii of convergence?

Common examples include: e^x has R = infinity; sin(x) and cos(x) have R = infinity; 1/(1-x) (geometric series) has R = 1; ln(1+x) has R = 1; the series sum of x^n/n! has R = infinity; and sum of n!*x^n has R = 0.

Additional Resources

• Radius of Convergence - Wikipedia
• Power Series - Wikipedia
• Ratio Test - Wikipedia
• Root Test - Wikipedia