On Which Interval Does The Power Series Convergence Calculator

Determining the interval of convergence for a power series is a fundamental concept in calculus and analysis. This calculator helps you find the interval where a given power series converges, using the ratio test and endpoint analysis.

What is a Power Series?

A power series is an infinite series of the form:

Σ (from n=0 to ∞) aₙ (x - c)ⁿ = a₀ + a₁(x - c) + a₂(x - c)² + a₃(x - c)³ + ...

Where:

aₙ are coefficients
x is the variable
c is the center of the series (often 0)

Power series are important in mathematics, physics, and engineering because they can represent functions as infinite sums of simpler terms.

Convergence Criteria

The interval of convergence is the set of all x-values for which the power series converges. To find it, we use the ratio test:

L = lim (n→∞) |aₙ₊₁ / aₙ|

The series converges absolutely when L < 1 and diverges when L > 1. When L = 1, we must test the endpoints.

Note: The ratio test only determines absolute convergence. Conditional convergence requires additional analysis.

How to Find the Interval of Convergence

Step 1: Apply the Ratio Test

Calculate the limit L using the ratio test formula. This gives you the radius of convergence (R).

R = lim (n→∞) |aₙ / aₙ₊₁|

The interval of potential convergence is (c - R, c + R).

Step 2: Test the Endpoints

Evaluate the series at x = c + R and x = c - R to determine if these points are included in the interval.

If the series converges at an endpoint, include it in the interval
If it diverges, exclude it

Step 3: Final Interval

The complete interval of convergence is the combination of the radius interval and the tested endpoints.

Examples

Example 1: Simple Power Series

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

Using the ratio test:

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

The series converges for all x, so the interval of convergence is (-∞, ∞).

Example 2: Series with Finite Interval

Consider the series Σ (from n=0 to ∞) (x - 2)ⁿ / 3ⁿ.

Using the ratio test:

L = lim (n→∞) |(x - 2)^(n+1) / 3^(n+1) / ((x - 2)ⁿ / 3ⁿ)| = |(x - 2)/3| = |x - 2|/3

The series converges when |x - 2|/3 < 1 → |x - 2| < 3 → -1 < x < 5.

Testing endpoints:

At x = -1: Σ (from n=0 to ∞) (-3)ⁿ / 3ⁿ = Σ (-1)ⁿ → Diverges
At x = 5: Σ (from n=0 to ∞) 3ⁿ / 3ⁿ = Σ 1 → Diverges

The interval of convergence is (-1, 5).

FAQ

What is the difference between radius and interval of convergence?: The radius of convergence is the distance from the center where the series converges. The interval of convergence includes the center and may include endpoints beyond the radius.
Why do we need to test the endpoints?: The ratio test only tells us about convergence within the radius. The endpoints may or may not be included, so we must test them separately.
Can a power series converge at only one point?: Yes, if the radius of convergence is zero and the series only converges at the center point.
What if the ratio test gives L = 1?: When L = 1, the ratio test is inconclusive. You must use other tests like the root test or direct comparison.
How do I know if a series converges conditionally?: Conditional convergence occurs when the series converges but not absolutely. You would need to check the convergence of the absolute series and compare it to the original series.