Interval of Convergence Calculator for Power Series

Determine the interval of convergence for a power series with our precise calculator and comprehensive guide. Understand the Radius of Convergence and perform Endpoint Analysis to find the exact range where the series converges.

What is Interval of Convergence?

The interval of convergence for a power series is the set of all real numbers x for which the series converges. It's determined by the Radius of Convergence and Endpoint Analysis.

Power series are infinite sums of the form:

General Power Series Form

f(x) = Σ (from n=0 to ∞) cₙ (x - a)ⁿ

Where cₙ are coefficients and a is the center of the series. The interval of convergence tells us where this series converges to a finite value.

How to Calculate Interval of Convergence

The process involves three main steps:

Find the Radius of Convergence (R)
Test the endpoints of the interval (-R, R)
Combine the results to form the interval of convergence

Note: The Radius of Convergence is the distance from the center a where the series converges. The interval of convergence is the range from a-R to a+R, adjusted by endpoint tests.

Radius of Convergence

The Radius of Convergence (R) is found using the Ratio Test:

Ratio Test Formula

lim (n→∞) |cₙ₊₁ / cₙ| = L

If L < 1, the series converges absolutely for all x within the radius.

R = lim (n→∞) |cₙ / cₙ₊₁|

If the limit L is less than 1, the series converges for all x within the radius R. If L is greater than 1, the series diverges everywhere.

Endpoint Analysis

After finding R, test the endpoints x = a + R and x = a - R to determine if they should be included in the interval of convergence.

Use the Limit Comparison Test or Direct Substitution to test each endpoint:

Endpoint Test

lim (n→∞) |cₙ (R)ⁿ| = L

If L is finite and not infinity, the endpoint is included.

Example Calculation

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

Find the Radius of Convergence using the Ratio Test:

lim (n→∞) |(x-3)ⁿ⁺¹ / (n+1)! / [(x-3)ⁿ / n!]| = lim |(x-3)| / (n+1) = |x-3|

Set |x-3| < 1 → -1 < x-3 < 1 → 2 < x < 4

Thus, R = 1
Test the endpoints:
- At x = 4: lim (n→∞) (4-3)ⁿ / n! = lim (1)ⁿ / n! = 0 → included
- At x = 2: lim (n→∞) (2-3)ⁿ / n! = lim (-1)ⁿ / n! → does not converge → excluded
Final interval of convergence: [3, 4)

FAQ

What is the difference between Radius of Convergence and Interval of Convergence?

The Radius of Convergence is the distance from the center where the series converges. The Interval of Convergence is the actual range of x-values where the series converges, which may be smaller than the radius if endpoints don't converge.

How do I know if a series converges at an endpoint?

Use the Limit Comparison Test or Direct Substitution to test the endpoint. If the limit is finite and not infinity, the endpoint is included in the interval of convergence.

What if the Ratio Test gives an indeterminate form?

If the Ratio Test results in an indeterminate form, try the Root Test or other convergence tests. The Root Test is often more effective when the Ratio Test fails.