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Radius of Convergence — Definition, Formula & Examples

Radius of Convergence

The distance between the center of a power series' interval of convergence and its endpoints. If the series only converges at a single point, the radius of convergence is 0. If the series converges over all real numbers, the radius of convergence is ∞.

See also

Convergence tests, power series convergence, Taylor series, Maclaurin series, infinity

Key Formula

R=1lim⁡n→∞∣an+1an∣R = \frac{1}{\displaystyle\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|}
Where:
  • RR = The radius of convergence
  • ana_n = The coefficient of the nth term of the power series
  • an+1a_{n+1} = The coefficient of the (n+1)th term of the power series

Worked Example

Problem: Find the radius of convergence of the power series ∑n=0∞xn3n\displaystyle\sum_{n=0}^{\infty} \frac{x^n}{3^n}.
Step 1: Identify the coefficients. Here the general term is xn3n\frac{x^n}{3^n}, so the coefficient is an=13na_n = \frac{1}{3^n}.
an=13na_n = \frac{1}{3^n}
Step 2: Apply the Ratio Test. Compute the limit of the absolute ratio of consecutive coefficients.
lim⁡n→∞∣an+1an∣=lim⁡n→∞1/3n+11/3n=lim⁡n→∞3n3n+1\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \frac{1/3^{n+1}}{1/3^n} = \lim_{n \to \infty} \frac{3^n}{3^{n+1}}
Step 3: Simplify the limit.
lim⁡n→∞3n3n+1=lim⁡n→∞13=13\lim_{n \to \infty} \frac{3^n}{3^{n+1}} = \lim_{n \to \infty} \frac{1}{3} = \frac{1}{3}
Step 4: Take the reciprocal to find the radius of convergence.
R=11/3=3R = \frac{1}{1/3} = 3
Answer: The radius of convergence is R=3R = 3. The series converges for ∣x∣<3|x| < 3 and diverges for ∣x∣>3|x| > 3.

Another Example

This example shows the edge case where the factorial in the coefficients causes the series to diverge everywhere except the center. It illustrates that not every power series converges on a meaningful interval.

Problem: Find the radius of convergence of the power series ∑n=0∞n!2nxn\displaystyle\sum_{n=0}^{\infty} \frac{n!}{2^n} x^n.
Step 1: Identify the coefficients: an=n!2na_n = \frac{n!}{2^n}.
an=n!2na_n = \frac{n!}{2^n}
Step 2: Compute the ratio of consecutive coefficients.
∣an+1an∣=(n+1)!/2n+1n!/2n=(n+1)2\left| \frac{a_{n+1}}{a_n} \right| = \frac{(n+1)!/2^{n+1}}{n!/2^n} = \frac{(n+1)}{2}
Step 3: Evaluate the limit as n→∞n \to \infty.
lim⁡n→∞n+12=∞\lim_{n \to \infty} \frac{n+1}{2} = \infty
Step 4: Since the limit is ∞\infty, the radius of convergence is the reciprocal: R=1∞=0R = \frac{1}{\infty} = 0.
R=0R = 0
Answer: The radius of convergence is R=0R = 0. This series converges only at x=0x = 0 and diverges for every other value of xx.

Frequently Asked Questions

What is the difference between radius of convergence and interval of convergence?
The radius of convergence RR is a single non-negative number (or ∞\infty) that tells you how far from the center the series converges. The interval of convergence is the actual set of xx-values for which the series converges. For a series centered at cc with radius RR, the interval is contained in (c−R, c+R)(c - R,\, c + R), but you must separately test each endpoint x=c−Rx = c - R and x=c+Rx = c + R to determine whether they are included.
Can you use the Root Test instead of the Ratio Test to find the radius of convergence?
Yes. The Root Test gives R=1lim⁡n→∞∣an∣1/nR = \frac{1}{\lim_{n \to \infty} |a_n|^{1/n}}. This is sometimes easier, especially when the coefficients involve nnth powers. Both methods yield the same radius when the limits exist.
What happens at xx-values exactly on the boundary of the radius of convergence?
At x=c+Rx = c + R and x=c−Rx = c - R, the series may converge or diverge — you cannot tell from the radius alone. You must substitute each endpoint into the series and apply a separate convergence test (such as the Alternating Series Test or the pp-series test) to decide.

Radius of Convergence vs. Interval of Convergence

Radius of ConvergenceInterval of Convergence
What it isA single number R≥0R \geq 0 (or ∞\infty)A set (interval) of xx-values
What it tells youHow far from the center the series convergesExactly which xx-values make the series converge
Endpoint informationDoes not determine convergence at endpointsIncludes or excludes endpoints based on testing
How to find itRatio Test or Root Test on the coefficientsFind RR first, then test each endpoint separately
ExampleR=3R = 3[−3,3)[-3, 3) or (−3,3](-3, 3] or (−3,3)(-3, 3) or [−3,3][-3, 3]

Why It Matters

Radius of convergence appears throughout Calculus II and beyond whenever you work with power series, Taylor series, or Maclaurin series. Knowing the radius tells you the domain on which a function can be reliably represented or approximated by its series expansion. It is also essential in physics and engineering, where series solutions to differential equations are only valid within the radius of convergence around the expansion point.

Common Mistakes

Mistake: Forgetting to check the endpoints after finding the radius of convergence.
Correction: The Ratio or Root Test is inconclusive when ∣x−c∣=R|x - c| = R. You must substitute each endpoint into the original series and apply a separate test (e.g., Alternating Series Test, comparison) to determine whether the interval of convergence is open, closed, or half-open.
Mistake: Applying the Ratio Test to the entire term anxna_n x^n and forgetting to isolate ∣x∣|x|.
Correction: When using the Ratio Test on a power series, the factor ∣x−c∣|x - c| should be pulled outside the limit. The series converges when ∣x−c∣⋅lim⁡∣an+1/an∣<1|x - c| \cdot \lim |a_{n+1}/a_n| < 1, which you then solve for ∣x−c∣<R|x - c| < R. Mixing up the algebra can give an incorrect radius.

Related Terms