Radius and Interval of Convergence

You have a power series, you need its interval of convergence, and every online calculator you tried either gives a radius without checking endpoints or demands a subscription for the full answer. The ratio test gets you the radius quickly, but the endpoint check is where most homework answers go wrong. The procedure is covered in Stewart section 11.8 and OpenStax Calculus Volume 2 section 6.1. Three worked examples and a checklist follow.

A power series is an infinite series of the form Σ c_n (x − a)ⁿ. The constant a is the center of the series, and the coefficients c_n are fixed numbers that do not depend on x. The series converges for some x values and diverges for others. The set of x where it converges is the interval of convergence, and half the length of that interval (not counting the endpoints) is the radius of convergence R. Every calculus student who has faced a power series problem from Stewart section 11.8 or OpenStax section 6.1 has had to find these two things.

The ratio test is the standard method to find R. For a power series Σ c_n (x − a)ⁿ, compute L = lim_n→∞ |c_n+1/c_n| · |x − a|. The series converges absolutely when L < 1 and diverges when L > 1. Solving the inequality L < 1 gives |x − a| < R, where R = 1 / lim_n→∞ |c_n+1/c_n|, provided the limit exists. The result is straightforward when the coefficients are rational or factorial. The root test is an alternative: compute ρ = limsup_n→∞ |c_n|^1/n, then R = 1/ρ. The root test is often easier when terms have nth powers.

The ratio test is inconclusive at the endpoints where |x − a| = R. At those two points, the series may converge absolutely, converge conditionally, or diverge. You must plug each endpoint into the original series and use a different convergence test, usually the p-series test, alternating series test, or comparison test, to determine what happens there. Stewart is explicit about this in section 11.8: the ratio test gives R, but only endpoint testing gives the interval. OpenStax section 6.1 follows the same procedure.

The Series

Find the interval of convergence for Σ_n=0^∞ (x − 2)ⁿ / (n+1).

Apply The Ratio Test

c_n = 1/(n+1). Then lim_n→∞ |c_n+1/c_n| = lim_n→∞ (n+1)/(n+2) = 1. So R = 1/1 = 1. The series converges when |x − 2| < 1, i.e., on the open interval (1, 3).

Test Endpoints

At x = 3, the series becomes Σ 1/(n+1), the harmonic series with p=1, which diverges. At x = 1, the series becomes Σ (−1)ⁿ/(n+1). This is an alternating series with terms decreasing to zero, so by the alternating series test it converges conditionally. The interval of convergence is [1, 3).

The Series

Find the radius and interval for Σ_n=0^∞ n! xⁿ.

Apply The Ratio Test

c_n = n!. Then lim_n→∞ |c_n+1/c_n| = lim_n→∞ (n+1) = ∞. So R = 1/∞ = 0. The series converges only when |x − 0| < 0, meaning only at x = 0. There are no endpoints to test because the interval is a single point {0}.

The Series

Find the radius and interval for Σ_n=0^∞ xⁿ / n!.

Apply The Ratio Test

c_n = 1/n!. Then lim_n→∞ |c_n+1/c_n| = lim_n→∞ 1/(n+1) = 0. So R = 1/0 = ∞. The series converges for all real x. The interval of convergence is (−∞, ∞). No endpoints to test.

When To Use The Root Test

The root test is an alternative to the ratio test. For a power series Σ c_n (x − a)ⁿ, compute ρ = limsup_n→∞ |c_n|^1/n. The radius R = 1/ρ. The root test is especially useful when the coefficients involve nth powers, such as c_n = (2n)ⁿ or c_n = (ln n)ⁿ. Both tests give the same R when both are applicable.

Cauchy-Hadamard Formula

The formal statement is the Cauchy-Hadamard theorem: R = 1 / limsup_n→∞ |c_n|^1/n. This formula works for any power series, even when the ratio test is inconclusive because the limit does not exist. In practice, for most textbook problems, the ratio test is simpler because it uses the familiar ratio of consecutive terms.

Follow these steps every time you find R from the ratio or root test. Failure at any one step means the interval is incomplete.

• Write down the two endpoint values: x = a − R and x = a + R.
• Substitute each into the original power series. The result is a numeric series (no x variable left).
• Determine convergence of each numeric series using a test appropriate to its form. Common choices: p-series test, alternating series test, comparison test, limit comparison test, or the integral test.
• Record whether each endpoint converges absolutely, converges conditionally, or diverges.
• State the interval of convergence using brackets for endpoints that converge and parentheses for endpoints that diverge.

An interval of convergence calculator that claims to find the full interval automatically is misleading. Current tools can compute the radius R by applying the ratio or root test, but they cannot reliably check endpoints because endpoint convergence depends on the specific series and the choice of test. A calculator that shows steps may handle simple rational series, but it will fail on series requiring case-by-case analysis, such as those with logarithmic coefficients or factorial terms that cancel in unexpected ways. Rely on the calculator only for the radius. Check endpoints yourself with the checklist above.

The most common mistake students make is stopping after finding R and writing the interval as (a − R, a + R) without testing the endpoints. A series with R = 1 can have an interval of [−1, 1], (−1, 1), [−1, 1), or be missing one or both endpoints entirely. The ratio test is silent on endpoints. Stewart section 11.8 and OpenStax section 6.1 both emphasise this point with examples where the endpoints behave differently. Always test them.