Interval of Convergence Calculator Symbolab

Determine the interval of convergence for power series using our Symbolab-powered calculator. This tool helps you find where a series converges and provides a visual representation of the convergence behavior.

What is Interval of Convergence?

The interval of convergence is the set of all real numbers x for which an infinite series converges. For a power series centered at a = 0, it's typically written as -R ≤ x ≤ R, where R is the radius of convergence.

Power series are infinite sums of the form:

Σ (from n=0 to ∞) cₙxⁿ = c₀ + c₁x + c₂x² + c₃x³ + ...

The interval of convergence depends on the coefficients cₙ and the behavior of the series as x approaches infinity.

How to Find Interval of Convergence

To find the interval of convergence, follow these steps:

Identify the power series and its coefficients cₙ.
Apply the Ratio Test to find the radius of convergence R.
Check the endpoints ±R to determine if they're included in the interval.
Combine the results to form the interval of convergence.

The Ratio Test 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=0 to ∞) (xⁿ)/n². Let's find its interval of convergence.

Apply the Ratio Test:
lim(n→∞) |(xⁿ⁺¹)/(n+1)²| / |(xⁿ)/n²| = lim(n→∞) |x|(n²)/(n+1)² = |x|
The series converges when |x| < 1, so R = 1.
Check the endpoints:
- At x = 1: The series becomes Σ1/n², which converges by the p-series test.
- At x = -1: The series becomes Σ(-1)ⁿ/n², which converges absolutely.
The interval of convergence is [-1, 1].

Interpretation of Results

The interval of convergence tells you:

Where the series converges absolutely (inside the interval)
Where the series may converge conditionally (at the endpoints)
Where the series definitely diverges (outside the interval)

For practical applications, you typically use the open interval (-R, R) for calculations, but the closed interval [-R, R] gives the complete range of convergence.

Frequently Asked Questions

What is the difference between radius and interval of convergence?

The radius of convergence (R) is the distance from the center of the series where the series converges. The interval of convergence includes the center ± R and any endpoints where the series might converge.

How do I know if a series converges at the endpoints?

You need to test the endpoints separately using other convergence tests like the nth-term test, comparison test, or integral test.

What if the Ratio Test gives L = 1?

When L = 1, the Ratio Test is inconclusive. You'll need to use another test like the Root Test or direct comparison.

Can a series converge for all x?

Yes, if the radius of convergence is infinite (R = ∞), the series converges for all real numbers x.