Radius of Convergence

The radius of convergence R is the half-length of the interval where a power series converges. Find R, and you know the series converges for all x within R units of the center c, and diverges for all x more than R units away. The critical fact: R tells you nothing about the endpoints x = c ± R. Test those separately. Most students who stop after finding R lose points on every exam.

For a power series ∑ a_n (x − c)ⁿ, the radius R is defined as the non-negative number such that the series converges when |x − c| < R and diverges when |x − c| > R. The ratio test and the root test are the two standard tools to compute R. Each test produces the same R when both limits exist, but they differ in ease of use depending on the form of a_n. The limit superior version of the root test (Cauchy, Hadamard) is the most general formula and works when the ratio test limit does not exist.

The radius of convergence is defined as R = 1 / lim sup |a_n|^1/n (Cauchy, Hadamard formula) or, when the simpler limit exists, R = lim |a_n / a_n+1|. This number is always non-negative. It can be 0, a positive finite number, or ∞. The open interval (c − R, c + R) is guaranteed to be part of the interval of convergence, but the actual interval may be smaller if endpoints diverge.

Stewart's Calculus (9th edition, Section 11.8, Theorem 3) and OpenStax Calculus Vol. 2 (2nd edition, Section 5.3) both give this definition. The center c is the point around which the series is expanded; shifting the center produces an entirely different series. For standard Maclaurin series (center 0), the radius is measured from 0.

Using the Ratio Test

The ratio test computes R from the limit L = lim_n→∞ |a_n+1 / a_n|. Then R = 1/L, with the conventions L = 0 giving R = ∞ and L = ∞ giving R = 0. This is the standard method in Stewart Section 11.6 and OpenStax Section 5.5. Use the ratio test when a_n contains factorials, because the ratio a_n+1/a_n simplifies cleanly.

Worked example 1: Find R for ∑ (n!)(x − 3)ⁿ. Here a_n = n!. Compute |a_n+1 / a_n| = (n+1)!/n! = n+1. As n → ∞, L = ∞, so R = 0. The series converges only at x = 3.

Worked example 2: Find R for ∑ (xⁿ)/n!. Here a_n = 1/n!. Ratio = 1/(n+1) → 0, so R = ∞. This is the series for e^x, which converges for all real x. OpenStax Table 6.3 confirms this.

Using the Root Test (Cauchy, Hadamard)

The root test uses L = lim_n→∞ |a_n|^1/n. Then R = 1/L, with the same conventions. Use the root test when a_n contains an exponent of n, such as a_n = (n² + 1)/2ⁿ. The root test often handles terms with powers more cleanly than the ratio test. Stewart Section 11.7 and OpenStax Section 5.6 cover this.

Worked example 3: Find R for ∑ ((2n + 1)/3ⁿ)(x + 1)ⁿ. Here a_n = (2n + 1)/3ⁿ. Apply the root test: |a_n|^1/n = ((2n + 1)^1/n)/3. As n → ∞, (2n + 1)^1/n → 1, so L = 1/3, and R = 3. The series converges for |x + 1| < 3, i.e., −4 < x < 2, pending endpoint checks.

Failure case: If a_n = 0 for some n, the ratio test fails because division by zero occurs. The root test with lim sup handles this. For example, a series with a_n = 0 for odd n and a_n = 1/2ⁿ for even n requires the root test. Stewart Section 11.7 notes this limitation.

The radius of convergence formula is R = lim_n→∞ |a_n / a_n+1| when the limit exists. If the limit does not exist, use the Cauchy, Hadamard formula: R = 1 / lim sup_n→∞ |a_n|^1/n. Both formulas give the same R when both limits exist. The Cauchy, Hadamard formula is the general definition, but most calculus courses teach the ratio test limit first. OpenStax Section 5.5 and Stewart Section 11.8 present the ratio version as the primary method.

The formula works only for power series of the form ∑ a_n (x − c)ⁿ. Applying it to a Fourier series or a numerical method series produces a meaningless result. For Taylor series, the coefficients a_n come from derivatives of the function at c. The radius of convergence for a Taylor series is the same as the radius of the power series, not separate.

Always Test the Endpoints

The radius of convergence is a single number (R). The interval of convergence is the set of x-values where the series converges, which may be (c − R, c + R), [c − R, c + R), (c − R, c + R], or [c − R, c + R]. The radius determines the length of the interval, but not its endpoints. The confusion pair is the most common mistake on calculus exams: students find R and then write (c − R, c + R) as the final answer without testing endpoints.

Worked example 4: For ∑ (xⁿ)/n, the ratio test gives R = 1. At x = 1, the series becomes ∑ 1/n, the harmonic series, which diverges. At x = −1, it becomes ∑ (−1)ⁿ/n, the alternating harmonic series, which converges conditionally. So the interval of convergence is [−1, 1), not (−1, 1). The radius (R = 1) is the same in both cases, but the interval differs.

The endpoint test for x = 1 uses the p-series test with p = 1. For x = −1, the alternating series test applies. Stewart Section 11.8 and OpenStax Section 5.5 both emphasize this separation of tasks. The ratio test is inconclusive at endpoints, so you must switch to p-series, alternating series, comparison, or integral tests there.

R = 0

The series converges only at x = c. This happens when the coefficients grow very fast, such as a_n = n! or a_n = nⁿ. The series is essentially useless for approximation away from the center. Example: ∑ n! xⁿ has R = 0. OpenStax Section 5.3 notes this case. If your series gives R = 0, consider a different expansion point or a non-power-series method.

R = ∞

The series converges for every real x. This occurs when coefficients decay super-exponentially, as in e^x (∑ xⁿ/n!), sin x, and cos x. OpenStax Table 6.3 lists these as entire functions. No endpoint testing is needed because there are no endpoints. Use the series freely over any domain, but note that convergence rate varies with distance from the center, closer to c gives faster convergence. The nth-term test is satisfied automatically for all x.

Failure case: A student who gets R = ∞ and assumes the series converges for all x but does not verify it is a power series will misapply this result to non-power series like Fourier series, where convergence behavior is different. Stick to power series of the form ∑ a_n (x − c)ⁿ.

The type of coefficient a_n determines R. Polynomial coefficients (a_n = n^k or polynomial in n) give R = 1, because the ratio test limit is 1. Factorial coefficients (a_n = n!) give R = 0. Exponential coefficients (a_n = 1/n!) give R = ∞. Rational coefficients with powers (a_n = (2n)/(3ⁿ)) give a finite positive R. This pattern appears in every calculus textbook: Stewart Section 11.8 exercises and OpenStax Section 5.3 examples.

Worked example 5: For ∑ (n² + 1) xⁿ, the ratio test gives lim |(n+1)² + 1| / |n² + 1| = 1, so R = 1. At endpoints, the series becomes ∑ (n² + 1) and ∑ (−1)ⁿ(n² + 1), both of which fail the nth-term test (terms do not approach zero), so the interval is (−1, 1).