Interval of Convergence Calculator Online

The interval of convergence is the set of all real numbers x for which a power series converges. Determining this interval is crucial in calculus and mathematical analysis. Our online calculator helps you find the interval of convergence for any given power series.

What is Interval of Convergence?

A power series is an infinite series of the form:

Σ (from n=0 to ∞) aₙ (x - c)ⁿ

The interval of convergence is the set of all x values for which this series converges. It's typically expressed as (a, b), where a and b are real numbers, possibly including infinity.

There are three possible scenarios for the interval of convergence:

Only x = c converges (the interval is empty)
The series converges only at x = c (radius of convergence is 0)
The series converges for all real x (radius of convergence is ∞)

How to Calculate Interval of Convergence

The standard method for determining the interval of convergence involves the ratio test:

lim (n→∞) |aₙ₊₁ / aₙ| = L

Where L is the limit of the ratio of consecutive coefficients. The radius of convergence R is then:

R = 1/L (if L ≠ 0)

Once you have the radius R, the interval of convergence is (c - R, c + R). You must then test the endpoints separately to determine if they are included in the interval.

Note: The ratio test may not work for all series. In such cases, you may need to use the root test or other convergence tests.

Examples

Example 1: Simple Power Series

Consider the series Σ (from n=0 to ∞) xⁿ / n!.

Using the ratio test:

lim (n→∞) |(xⁿ⁺¹ / (n+1)!) / (xⁿ / n!)| = lim (n→∞) |x / (n+1)| = 0

Since the limit is 0 for all x, the series converges for all real numbers. Therefore, the interval of convergence is (-∞, ∞).

Example 2: Series with Finite Radius

Consider the series Σ (from n=0 to ∞) (-1)ⁿ x²ⁿ / n.

Using the ratio test:

lim (n→∞) |[(-1)ⁿ⁺¹ x²ⁿ⁺¹ / (n+1)] / [(-1)ⁿ x²ⁿ / n]| = lim (n→∞) |x² (n / (n+1))| = |x²|

The radius of convergence is R = 1/|x²|, but this doesn't make sense in this context. Instead, we should recognize this as a geometric series with ratio x². The series converges when |x²| < 1, or |x| < 1. Therefore, the interval of convergence is (-1, 1).

FAQ

What is the difference between radius of convergence and interval of convergence?: The radius of convergence is the distance from the center c where the series converges. The interval of convergence includes the radius and may include the endpoints.
How do I know if the endpoints are included in the interval of convergence?: You must test the endpoints separately using substitution or other convergence tests. The series may converge at one endpoint but not the other.
What if the ratio test gives an indeterminate form?: If the limit L is 1, the ratio test is inconclusive. In this case, you may need to use the root test or other methods to determine convergence.
Can a power series converge for all x except one point?: Yes, but this is rare. The interval of convergence is typically an interval (a, b) or a single point, not a collection of isolated points.