Radius and Interval of Convergence
A power series is an infinite series of the form ∑aₙ(x − c)ⁿ, where c is the center and the aₙ are coefficients. Unlike a finite polynomial, a power series only makes sense — only converges to a finite value — for certain values of x. The set of all x for which the series converges is called the interval of convergence, and the distance from the center to the edge of that interval is called the radius of convergence R. Every power series falls into exactly one of three cases: it converges only at the single point x = c (R = 0), it converges for all real numbers (R = ∞), or it converges on a finite open interval (c − R, c + R) and diverges outside it. The endpoints x = c − R and x = c + R must be checked individually, because the series may converge, diverge, or exhibit conditional convergence there — the general theory cannot decide for us. Finding R is an essential skill in calculus and analysis because it tells you exactly where you can replace a complicated function with its power series representation. Functions like eˣ, sin x, and ln(1 + x) each have power series whose radii of convergence determine where those series are valid substitutes — and understanding this is the gateway to advanced topics such as Taylor series, differential equations, and complex analysis.
- Write the general term
- Apply the Ratio Test
- Solve for R
- Check the endpoints individually
- State the interval of convergence
Write the general term
Identify the general term aₙ of your power series ∑aₙ(x − c)ⁿ. For example, for the series ∑(xⁿ/n), the general term is aₙ = 1/n, so the full nth term is (1/n)xⁿ.
Apply the Ratio Test
The most reliable method is the Ratio Test. Compute L = lim(n→∞) |a_{n+1}(x−c)^{n+1} / (aₙ(x−c)ⁿ)| = |x − c| · lim(n→∞) |a_{n+1}/aₙ|. The series converges absolutely when L < 1 and diverges when L > 1.
Solve for R
Let ρ = lim(n→∞) |a_{n+1}/aₙ| (or use the Root Test: ρ = lim(n→∞) |aₙ|^{1/n}). Then R = 1/ρ, provided ρ ≠ 0 and ρ ≠ ∞. If ρ = 0 then R = ∞; if ρ = ∞ then R = 0. The series converges absolutely for all x with |x − c| < R.
Check the endpoints individually
Substitute x = c + R and x = c − R into the original series. Use other convergence tests — the Divergence Test, Alternating Series Test, p-series, or Comparison Test — to determine whether the series converges or diverges at each endpoint.
State the interval of convergence
Write the interval of convergence using the appropriate bracket notation: use a parenthesis ( ) at an endpoint if the series diverges there (open), and a square bracket [ ] if it converges there (closed). For example: [c − R, c + R), or (c − R, c + R), or [c − R, c + R].
The Key Equation
- R — Radius of convergence (unitless): Half-length of the open interval on which the power series converges absolutely. R = 0 means convergence at center only; R = ∞ means convergence everywhere.
- a_n — nth coefficient (unitless): The coefficient of the (x − c)ⁿ term in the power series ∑aₙ(x − c)ⁿ.
- a_{n+1} — (n+1)th coefficient (unitless): The coefficient of the next term; the ratio |a_{n+1}/aₙ| measures how rapidly the coefficients grow or shrink.
- c — Center of the series (unitless): The value of x around which the power series is expanded; convergence is symmetric about c within radius R.
Worked Example
Problem. Find the radius and interval of convergence of the power series ∑(n=1 to ∞) (−1)ⁿ xⁿ / n.
Answer. Radius of convergence R = 1; interval of convergence is (−1, 1].
Origins · 1 minAugustin-Louis Cauchy and how this was discovered
Cauchy placed the theory of infinite series on a rigorous foundation, introducing formal definitions of convergence and divergence. He proved the Cauchy-Hadamard theorem, which gives the radius of convergence of a power series in terms of R = 1 / lim sup |aₙ|^{1/n}, and developed the Root Test and Ratio Test that students still use today. Cauchy published over 800 mathematical papers and books — more than almost any other mathematician in history. His collected works fill 27 volumes!
Related Concepts
- Taylor Series — Taylor series are the most important examples of power series; understanding radius of convergence tells you where a Taylor series accurately represents its function.
- Ratio Test — The Ratio Test is the primary tool used to compute the radius of convergence of a power series.
- Alternating Series Test — Often needed to determine convergence or divergence of a power series at its endpoints, where the series may become alternating.
- Absolute and Conditional Convergence — Inside its radius of convergence, a power series converges absolutely; at the endpoints, it may only converge conditionally — a key distinction.