Radius of Convergence Calculator

Math Calculus • Infinite Series and Sequences

Written by STEM Calculators Team Published December 30, 2025 Updated February 24, 2026

5. Radius Of Convergence Calculator

Estimates the radius of convergence \(R\) for \(\sum c_n (x-a)^n\) using the ratio/root tests (limsup sampling), checks endpoints \(a\pm R\), and visualizes the convergence disk in the complex plane.

Inputs

Coefficient \(c_n\) (function of \(n\) only)

Series form: \(\sum_{n=n_0}^{\infty} c_n (x-a)^n\). Enter \(c_n\) using variable n. Constants: pi, e. Functions: sin cos tan, sqrt abs exp ln/log, and fact(n). Use ^ for powers. Implicit multiplication like 2n is allowed.

Start index \(n_0\)

Center \(a\)

Method

Uses tail sampling to approximate a limsup-style limit.

Max \(n\) for estimation

Samples \(c_n\) up to this index to estimate the limit.

Tail size

Uses the last tail samples to compute a robust median / spread.

Options

Test endpoints \(a\pm R\) Show steps Auto-fit graph after solve Legend Mark \(a\pm R\) Show test point \(z_0\)

Test point \(z_0\) (complex)

\(\Re(z_0)\)

\(\Im(z_0)\)

Terms for numeric test

Used for endpoint and boundary-point heuristics.

Presets

Click a preset to load & evaluate.

Ready

Graph (complex plane)

Drag to pan • wheel/pinch to zoom. Shows the convergence disk \(|z-a|<R\).

x: 0, y: 0, zoom: 70

Results & steps

Enter \(c_n\), \(n_0\), and \(a\), then click Calculate.

Rate this calculator

0.0 /5 (0 ratings)

Be the first to rate.

Your rating

Name (optional) Review (optional)

You can update your rating any time.

5. Radius Of Convergence Calculator — Theory

Power series and radius of convergence

A power series centered at \(a\) has the form \[ \sum_{n=n_0}^{\infty} c_n (x-a)^n. \] There exists \(R \in [0,\infty]\) such that: the series converges absolutely for \(|x-a|R\). The endpoints \(x=a\pm R\) must be tested separately.

Ratio test form (limsup)

Define \[ L = \limsup_{n\to\infty}\left|\frac{c_{n+1}}{c_n}\right|. \] Then \[ \limsup_{n\to\infty}\left|\frac{c_{n+1}(x-a)^{n+1}}{c_n(x-a)^n}\right| = L|x-a|. \] The series converges absolutely when \(L|x-a|<1\), i.e. \[ |x-a| < \frac{1}{L} = R. \] Special cases: if \(L=0\) then \(R=\infty\); if \(L=\infty\) then \(R=0\).

Root test form (limsup)

Define \[ L = \limsup_{n\to\infty}|c_n|^{1/n}. \] Since \[ \limsup_{n\to\infty}|c_n(x-a)^n|^{1/n} = L|x-a|, \] the same rule applies: the series converges absolutely when \(L|x-a|<1\), hence \[ R=\frac{1}{L}. \]

Sample: \(\sum n!\,x^n\)

Here \(c_n=n!\). The ratio is \[ \left|\frac{c_{n+1}}{c_n}\right| = \frac{(n+1)!}{n!} = n+1 \to \infty. \] Thus \(L=\infty\), so \(R=0\). The series converges only at \(x=0\).

University note: singularities and analytic continuation

For many analytic functions, \(R\) equals the distance from \(a\) to the nearest complex singularity of the function represented by the power series. This links radius-of-convergence questions to complex analysis and singularity structure. (For Laurent series, one gets an annulus of convergence rather than a disk.)

Frequently Asked Questions

What is the radius of convergence for a power series?

For sum c_n (x-a)^n, the radius of convergence R is the distance from the center a within which the series converges absolutely. The series typically diverges for |x-a|>R, while behavior at |x-a|=R depends on the endpoints.

How is R computed using the ratio test or root test?

Using limsup forms, L = limsup |c_(n+1)/c_n| or L = limsup |c_n|^(1/n), and the radius is R = 1/L. Special cases include L=0 giving R=infinity and L=infinity giving R=0.

Why does the calculator test the endpoints a-R and a+R separately?

Convergence at the boundary |x-a|=R cannot be decided by the ratio or root test alone. Each endpoint can converge or diverge depending on the specific coefficients c_n.

What does the complex test point z0 show on the graph?

z0 is a point in the complex plane plotted relative to the disk |z-a|<R. It helps visualize whether a chosen point lies inside the estimated convergence region or near the boundary.

Rate this calculator

Frequently Asked Questions

Related calculators