Interval of Convergence Online Calculator

What is Interval of Convergence?

A power series is an infinite series of the form:

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 can be finite or infinite.

There are three possible outcomes for the interval of convergence:

1. Only the center point c converges (interval is {c})
2. All real numbers converge (interval is (-∞, ∞))
3. A finite interval (a, b) where a ≤ c ≤ b

How to Find Interval of Convergence

The most common method to find the interval of convergence is the ratio test. Here's the step-by-step process:

1. Write the power series in summation form
2. Apply the ratio test to find the limit L = lim(n→∞) |aₙ₊₁ / aₙ|
3. Determine the radius of convergence R:
   • If L < 1, R = 1/L
   • If L ≥ 1, R = 0
4. Check the endpoints c ± R to determine if they should be included

Example Calculation

Let's find the interval of convergence for the series:

Using the ratio test:

1. Identify aₙ = (x² - 4)ⁿ / (n + 1)²
2. Compute the limit: L = lim(n→∞) |(x² - 4)ⁿ⁺¹ / (n + 2)² × (n + 1)² / (x² - 4)ⁿ|
3. Simplify to L = |x² - 4|
4. Set L < 1: |x² - 4| < 1 → -1 < x² - 4 < 1 → 3 < x² < 5
5. Solve for x: -√5 < x < -√3 or √3 < x < √5
6. Check endpoints:
   • At x = √5: series diverges
   • At x = -√5: series diverges
   • At x = √3: series converges
   • At x = -√3: series converges

The interval of convergence is (-√5, -√3] ∪ [√3, √5).

Common Pitfalls

When working with interval of convergence, be aware of these common mistakes:

1. Assuming the series converges for all x: Power series often have limited intervals of convergence
2. Forgetting to check the endpoints: The radius of convergence only tells part of the story
3. Incorrectly applying the ratio test: Remember to take the absolute value and consider the limit
4. Misinterpreting the result: The interval may include or exclude endpoints based on convergence at those points