Wolfram Alpha Interval of Convergence Calculator

This calculator uses Wolfram Alpha's computational engine to determine the interval of convergence for power series. The interval of convergence is the set of all real numbers for which a power series converges.

What is Interval of Convergence?

The interval of convergence for a power series is the range of x-values for which the series converges. It's determined by analyzing the behavior of the series as x approaches infinity and negative infinity.

For a power series centered at a = 0:

Σ (from n=0 to ∞) cₙxⁿ

The interval of convergence can be:

An open interval (a, b)
A closed interval [a, b]
An infinite interval (a, ∞)
A single point {a}

How to Calculate Interval of Convergence

The standard method for finding the interval of convergence involves three steps:

Find the radius of convergence using the ratio test
Check for convergence at the endpoints
Combine the results to determine the interval

The ratio test is the most common method for determining the radius of convergence. It states that for a series Σaₙ, if lim(n→∞) |aₙ₊₁/aₙ| = L, then:

If L < 1, the series converges absolutely
If L > 1, the series diverges
If L = 1, the test is inconclusive

Example Calculation

Consider the series Σ (from n=1 to ∞) (xⁿ)/n³. We'll find its interval of convergence.

Step 1: Apply the Ratio Test

Let aₙ = xⁿ/n³. Then:

lim(n→∞) |aₙ₊₁/aₙ| = lim(n→∞) |(xⁿ⁺¹)/(n+1)³| / |(xⁿ)/n³| = lim(n→∞) |x| * (n³)/(n+1)³ = |x|

The series converges when |x| < 1, so the radius of convergence is R = 1.

Step 2: Check Endpoints

At x = 1:

Σ (from n=1 to ∞) 1/n³ converges (p-series with p=3 > 1)

At x = -1:

Σ (from n=1 to ∞) (-1)ⁿ/n³ converges absolutely

Step 3: Determine Interval

The series converges for all x in the closed interval [-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 of the series to the nearest point where the series diverges. The interval of convergence includes all points within the radius where the series converges, plus possibly the endpoints.
Can a power series have an infinite radius of convergence?: Yes, if the series converges for all real numbers, the radius of convergence is infinite, and the interval of convergence is (-∞, ∞).
How do I know if a series converges at an endpoint?: You need to test the series at each endpoint separately using other convergence tests like the nth term test, comparison test, or integral test.
What if the ratio test gives L = 1?: If the ratio test is inconclusive (L = 1), you may need to use other tests like the root test or the nth term test to determine convergence.