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Radius of Convergence Calculator
Last updated: May 31, 2026
Written by Blake Boege
A radius of convergence calculator is a mathematical utility designed to determine the radius R of a power series, which represents the distance from the center point within which the series converges absolutely. The calculator applies the Ratio Test by evaluating the limit of the absolute ratio of consecutive coefficients as n approaches infinity. It outputs whether the radius is a finite number, zero, or infinite. Students, mathematicians, and physics researchers use this tool to determine the domain of validity for power series expansions and Taylor approximations.
Calculate the radius of convergence R of a power series. Enter the general term, specify the center, and analyze the limit of terms step-by-step.
Quick Answer
Compute the radius of convergence R of a power series. Enter the general term to find the range of x-values where the series converges.
Use standard variable n. You may optionally include the variable x (e.g. x^n/n).
Try Examples
Numerical Ratio Test Trend
| n | |a_n| | |a_n₊₁| | |a_n / a_n₊₁| |
|---|---|---|---|
| 10 | 2.755732E-7 | 2.505211E-8 | 11 |
| 50 | 3.287949E-65 | 6.44696E-67 | 51 |
| 100 | 1.07151E-158 | 1.060901E-160 | 101 |
| 150 | 1.750276E-263 | 1.159123E-265 | 151 |
Radius R =
∞
Open Interval of Convergence: (-∞, ∞)
Ratio Test: R = lim |a_n / a_{n+1}|. If R = ∞, the series converges for all x. If R = 0, it converges only at the center point.
Note: Endpoint convergence (at center ± R) must be checked separately using other convergence tests (e.g. comparison or alternating series test).
Examples
a_n = 1/n!
R = ∞ (converges everywhere)
a_n = x^n/n
R = 1 (converges on [-1, 1))
a_n = n^2/3^n
R = 3 (converges on (-3, 3))
How it works
The calculator evaluates the ratio of consecutive coefficients of the power series at large values of n to estimate the limit:
Ratio Test Limit Formula
R = lim (n → ∞) |a_n / a_n₊₁|
Endpoint Convergence
The Ratio Test is inconclusive when the limit equals 1. In the context of a power series, this corresponds to the boundary points of the interval: $x = c - R$ and $x = c + R$. To determine if the interval is open, closed, or half-open, you must substitute these endpoint values back into the original power series and use other tests (like the Alternating Series Test or p-Series Test) to check for convergence.
Related Mathematical Solvers
If you want to determine whether the endpoints are included in the interval, visit our dedicated Interval of Convergence Calculator. For Taylor expansions, check out the Taylor Series Calculator.
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Frequently asked questions
The radius of convergence R of a power series is a non-negative number (or infinity) such that the series converges absolutely for |x - c| < R and diverges for |x - c| > R, where c is the center of the series.
By the Ratio Test, the series converges if the limit L = lim |a_{n+1} / a_n| * |x - c| < 1. Isolating |x - c| gives |x - c| < 1/L, so the radius of convergence R is equal to 1/L = lim |a_n / a_{n+1}|.
A radius of R = ∞ means that the power series converges for all real numbers x. The interval of convergence is (-∞, ∞). An example is the series for e^x, sin(x), or cos(x).
A radius of R = 0 means that the series only converges at its center point x = c. For all other values of x, the terms of the series grow too fast, causing the series to diverge.
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