Radius of Convergence and Interval Calculator

The radius of convergence is a fundamental concept in the theory of power series, determining the range of values for which the series converges. This calculator helps you determine both the radius and the interval of convergence for a given power series.

What is Radius of Convergence?

The radius of convergence (R) of a power series is the distance from the center of the series to the point where the series stops converging. For a power series centered at a = 0:

Σ (from n=0 to ∞) cₙxⁿ

The radius of convergence R is defined as the supremum of all x for which the series converges. The series converges absolutely for all x with |x| < R and diverges for all x with |x| > R.

There are three possible cases for the radius of convergence:

The series converges only at x = 0 (R = 0).
The series converges for all real x (R = ∞).
The series converges for all x in the interval (-R, R) and diverges elsewhere.

How to Calculate Radius of Convergence

The most common method to find the radius of convergence is the Ratio Test. For a power series Σ cₙxⁿ, the ratio test involves taking the limit:

L = lim (n→∞) |cₙ₊₁ / cₙ|

Based on the value of L, we can determine the radius of convergence:

If L < 1, the series converges for all x (R = ∞).
If L > 1, the series diverges for all x (R = 0).
If L = 1, the test is inconclusive.

If the Ratio Test is inconclusive (L = 1), other methods such as the Root Test or direct comparison may be used.

Note: The Ratio Test is not always conclusive. In some cases, the limit may not exist or may be equal to 1, requiring additional analysis.

Interval of Convergence

The interval of convergence is the set of all x values for which the power series converges. It is determined by the radius of convergence and additional endpoint tests.

Once the radius R is found, we need to check the endpoints x = R and x = -R to determine if they are included in the interval of convergence. This is typically done using substitution and limit analysis.

For example, if the series Σ (n=0 to ∞) (-1)ⁿxⁿ/2ⁿ has a radius of convergence R = 2, we would check:

At x = 2: Does Σ (-1)ⁿ2ⁿ/2ⁿ converge?
At x = -2: Does Σ (-1)ⁿ(-2)ⁿ/2ⁿ converge?

The interval of convergence would then be (-2, 2], (-2, 2), or [-2, 2] depending on the results of these endpoint tests.

Examples

Example 1: Simple Power Series

Consider the power series Σ (n=0 to ∞) xⁿ/n!. Let's find its radius of convergence.

Using the Ratio Test:

L = lim (n→∞) |x| / (n+1) = 0

Since L = 0 < 1, the series converges for all x. Therefore, the radius of convergence is R = ∞.

Example 2: Alternating Series

Consider the power series Σ (n=0 to ∞) (-1)ⁿxⁿ. Let's find its radius of convergence.

Using the Ratio Test:

L = lim (n→∞) |x| = |x|

The test is inconclusive when L = 1, so we use the Root Test:

L = lim (n→∞) |(-1)ⁿxⁿ|^(1/n) = |x|

Again, the test is inconclusive when L = 1. Therefore, we can conclude that the series converges only at x = 0, so R = 0.

FAQ

What is the difference between radius of convergence and interval of convergence?

The radius of convergence is the distance from the center of the power series to the point where the series stops converging. The interval of convergence is the set of all x values for which the series converges, which includes the radius and any additional endpoints that may be included.

How do I know if a power series converges at its radius?

To determine if a power series converges at its radius, you need to perform endpoint tests. These involve substituting the radius into the series and checking for convergence using methods like substitution, limit comparison, or integral tests.

Can the radius of convergence be negative?

No, the radius of convergence is always a non-negative real number. It represents a distance and cannot be negative.

What if the Ratio Test gives L = 1?

If the Ratio Test gives L = 1, the test is inconclusive. In this case, you may need to use other tests like the Root Test, direct comparison, or integral tests to determine the radius of convergence.