What Is the Interval of Convergence?

When you first encounter a power series—an infinite sum like Σ aₙ(x − c)ⁿ—one of the most important questions is: for which values of x does this series actually add up to a finite number? The answer is called the interval of convergence. In simple terms, the interval of convergence is the set of all x values that make the series converge (produce a finite sum). Outside that interval, the series diverges (grows without bound or fails to settle). Understanding this concept is essential for using power series in calculus, physics, and engineering.

Every power series has a center point c and a radius R that defines how far from c the series converges. The interval of convergence is either an open interval (c − R, c + R), or it may include one or both endpoints depending on the series. For example, if the series is centered at 0 and has a radius of 2, the series converges for x between −2 and 2, and possibly at the endpoints −2 or 2 if tests show convergence there.

What Is Interval of Convergence?

The interval of convergence is the complete set of x values for which the power series Σ aₙ(x − c)ⁿ converges. The series has a radius of convergence R (which can be a nonnegative number or infinity). If R = 0, the series converges only at the center x = c. If R = ∞, it converges for all real x. If R is a finite positive number, the series converges for |x − c| < R and diverges for |x − c| > R. The endpoints x = c ± R must be checked separately because convergence at the boundaries is not guaranteed by the ratio or root test.

For more on the definition and how it relates to the radius, see our page on What Different Radius and Interval Values Mean.

How Is the Interval of Convergence Calculated?

To find the interval of convergence, you first find the radius R using either the ratio test or the root test. Then you test the endpoints. The ratio test involves calculating L = lim (n → ∞) |aₙ₊₁ / aₙ| and then R = 1/L. The root test uses L = lim (n → ∞) √[n]{|aₙ|} and R = 1/L. If L = 0, R = ∞; if L = ∞, R = 0.

Once R is known, the series converges for |x − c| < R. Then you substitute the endpoints x = c + R and x = c − R into the original series and apply convergence tests (like the p-test, alternating series test, or direct comparison) to decide whether those points are included.

If you want a step-by-step walkthrough, check out our guide on How to Calculate Interval of Convergence: Step-by-Step Guide (2026). For the exact formulas, see the Interval of Convergence Formula: Ratio and Root Test Explained (2026).

A Simple Worked Example

Let's find the interval of convergence for the power series Σ (n!)(x − 0)ⁿ from n=0 to ∞. Here aₙ = n!, c = 0.

Step 1: Ratio test. Compute L = lim (n→∞) |aₙ₊₁ / aₙ| = lim (n→∞) |(n+1)! / n!| = lim (n→∞) (n+1) = ∞. Since L = ∞, R = 1/∞ = 0. So the series converges only when |x - 0| < 0, which means only at x = 0.

Interval of convergence: {0} (just the single point). The radius is 0.

Now try a different series: Σ (x − 2)ⁿ / (n³ + 1) from n=0 to ∞, centered at c = 2.

Step 1: Ratio test. aₙ = 1/(n³+1). Then |aₙ₊₁ / aₙ| = ( (n³+1) / ((n+1)³+1) ). As n→∞, this ratio approaches 1, so L = 1, and R = 1. So the series converges for |x − 2| < 1, i.e., 1 < x < 3.

Step 2: Test endpoints. At x = 1, the series becomes Σ (−1)ⁿ/(n³+1). This is an alternating series with decreasing terms that approach 0, so it converges (conditional convergence). At x = 3, the series becomes Σ (1)ⁿ/(n³+1) = Σ 1/(n³+1). This is like a p-series with p = 3 > 1, so it converges absolutely. Thus both endpoints are included.

Final interval: [1, 3] (closed interval).

Why Does the Interval of Convergence Matter?

The interval of convergence is crucial in calculus because power series are used to represent functions. For instance, the function 1/(1−x) can be written as Σ xⁿ for |x| < 1. Knowing the interval tells you where the series representation is valid. Outside that interval, the series diverges and cannot be used to approximate the function.

In Taylor and Maclaurin series, the interval of convergence determines where the polynomial approximation accurately matches the original function. For example, the Maclaurin series for e^x converges for all real x (R = ∞), while the series for ln(1+x) converges only for |x| < 1 (plus an endpoint). If you try to use the series outside its interval, you'll get false results. See our page on Interval of Convergence for Taylor and Maclaurin Series (2026) for more details.

Common Misconceptions

• Misconception: The ratio test gives the complete interval.
  Fact: The ratio test only gives the radius. Endpoints must be tested separately because the test fails when L = 1.
• Misconception: If the radius is finite, the interval is always open.
  Fact: The interval can be open, half-open, or closed depending on endpoints. For example, Σ xⁿ/n converges at x = −1 but diverges at x = 1 (interval [−1,1)).
• Misconception: A series with aₙ = 0 for all n has no interval.
  Fact: It converges for all x (interval (−∞, ∞)).

For more answers to common questions, visit our Interval of Convergence FAQ (2026).