Radius of Convergence Calculator

Calculate the radius of convergence of a power series using the Ratio or Root Test.
Enter the general term aₙ in terms of n.

General Term (aₙ):

Choose Method:

Test near n =

Formulas:
Ratio Test: $ R = \dfrac{1}{\lim_{n \to \infty} |a_{n+1}/a_n|} $
Root Test: $ R = \dfrac{1}{\lim_{n \to \infty} \sqrt[n]{|a_n|}} $

🧮 Radius of Convergence Calculator

The Radius of Convergence Calculator is a free online mathematics tool designed to find the radius of convergence of a power series quickly and accurately.
It helps students, teachers, and professionals determine where a given infinite series converges or diverges using either the Ratio Test or the Root Test.

In mathematical terms, the radius of convergence (R) defines the distance from the center of a power series where the series remains convergent.
This calculator automates that process, saving you time and reducing errors compared to manual calculations.

Whether you’re studying calculus, preparing for competitive exams, or working with infinite series in engineering or physics, this radius of convergence calculator online offers instant, reliable results.

For more useful academic and mathematical tools, explore our Math & Science Calculators section.

🧭 How to Use the Radius of Convergence Calculator

Using this calculator is straightforward and takes less than a minute:

Enter the general term of your series $a_n$ — e.g. 1/n^2, (3^n)/(n!), or (2^n)/(n^3).
Select the test type — choose between the Ratio Test or the Root Test.
Set the test value of n (default is 50, which approximates the limit as $\to \infty$ ).
Click Calculate Radius — the tool instantly computes:
- The limit value (L), and
- The radius of convergence (R = 1/L)
Review the result displayed with step-by-step symbolic math using MathJax for beautiful, accurate formulas.

This makes it perfect for homework, class demonstrations, or quick checks during study sessions.

You can also go through - Fourier Transform Calculator

⚙️ How the Radius of Convergence Calculator Works

The Radius of Convergence Calculator applies the most common convergence tests used in calculus and real analysis:

🧩 1. Ratio Test

For a series $∑an(x−c)n\sum a_n (x – c)^n$ , the radius of convergence is found by:

$\frac{1}{\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|}$

If $L = 0$ , then $\infty$ → the series converges for all $x$ .
If $\infty$ , then $R = 0$ → the series converges only at the center $x = c$ .

🧮 2. Root Test

Alternatively, using the Root Test:

$\frac{1}{\lim_{n \to \infty} \sqrt[n]{|a_n|}}$

This method is especially useful when each term contains exponential or factorial expressions.

Our calculator uses Math.js to process symbolic expressions and MathJax to render formulas clearly, giving users a combination of precision and readability.

🌟 Why Use This Radius of Convergence Calculator?

There are many reasons why CalcIreland’s Radius of Convergence Calculator is the ideal choice for students and professionals in Ireland:

✅ 1. Instant and Accurate Results

Get step-by-step results using proven mathematical formulas — no manual computation needed.

✅ 2. Supports Both Ratio and Root Tests

You can freely switch between methods based on the form of your power series.

✅ 3. Clean, Mobile-Friendly Interface

Optimized for fast use on desktops, tablets, and smartphones.

✅ 4. Free and Accessible Anytime

A 100% free online convergence calculator that doesn’t require registration or downloads.

✅ 5. Educational and SEO-Enhanced

Includes detailed explanations, worked examples, and clear mathematical notation — perfect for learning and teaching.

📘 Example Calculations

Series $a_n$	Method	Limit $L$	Radius $R = 1/ L$	Convergence
$1/n^2$	Ratio	0	∞	Convergent everywhere
$3^n)/(n!)$	Ratio	0	∞	Convergent everywhere
$2^n)/(n^3)$	Ratio	2	0.5	Convergent when
$5^n$	Root	5	0.2	Convergent when

❓ Frequently Asked Questions (FAQ)

1. What is a radius of convergence?

The radius of convergence $R$ is the distance from the center $c$ of a power series within which the series converges absolutely. Outside this radius, it diverges.

2. What does it mean if the radius of convergence is infinite?

It means the series converges for all values of $x$ , such as with exponential series like $e^x$ .

3. Which test should I choose — Ratio or Root Test?

Use the Ratio Test for factorials, exponentials, or products like $3^n)/(n!)$ .
Use the Root Test for expressions with powers like $n^k$ or $x/n)^n$ .

4. Can I use this calculator for complex numbers?

Yes. The calculator can evaluate expressions involving complex numbers as long as they are entered in standard form, such as (1+i)^n/n!.

5. Who can use this tool?

This tool is designed for students, math educators, and professionals studying calculus, infinite series, or real and complex analysis.