Radius of Convergence Calculator

For a power series ∑n=n₀∞ an(x − c)n, the radius is R = 1 / limsup(|an|^(1/n)). Enter either an explicit formula for a(n) or a list of coefficients.

Use n as the index. Examples: 1, 1/factorial(n), (3^n)/n^2, n!, 1/(2^n). If provided, this takes priority over the CSV below.
Comma-separated an starting at n = n₀ (below). Trailing zeros imply a polynomial (then R = ∞).
Series is in powers of (x − c). Default 0.
Usually 0 or 1. CSV starts at this index.
One letter, used in the inequality (e.g., x).
Click an example to auto-fill and see the result immediately.
Shows: power series form and the convergence condition |x − c| < R, with R simplified numerically (not LaTeX).

Results

Radius (R)

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Computed from root/ratio test estimates.

Convergence Condition

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Open interval around the center when R is finite.

Method & Stats

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Displays root/ratio estimates based on the tail of the sequence.
Helping notes:
  • If only finitely many non-zero coefficients (a polynomial), then R = ∞ (converges for all real x).
  • Root test estimate uses R ≈ 1 / median(|an|^(1/n)) over large n; ratio test uses R ≈ median(|an/an+1|).
  • If estimates disagree strongly, the calculator reports both and chooses the more stable one.
  • Endpoints (|x − c| = R) require separate testing and are not decided by the radius alone.

What is a Radius of Convergence Calculator?

A Radius of Convergence Calculator determines where a power series converges. For a series centered at \(x_0\), $$\sum_{n=0}^{\infty} a_n\,(x-x_0)^n,$$ the radius \(R\) specifies all \(x\) such that \(|x-x_0| < R\) yields convergence, while \(|x-x_0| > R\) yields divergence. The boundary points \(x=x_0\pm R\) must be checked separately. Two equivalent characterizations are the root and ratio tests:

$$\textbf{Root test:}\quad C=\limsup_{n\to\infty}\sqrt[n]{|a_n|},\qquad R=\frac{1}{C}\ (\text{with } 1/\infty=0,\ 1/0=\infty).$$ $$\textbf{Ratio test:}\quad L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|,\qquad R=\frac{1}{L}.$$

Inside the interval \((x_0-R,\ x_0+R)\) the series converges absolutely. At endpoints, convergence depends on the resulting series after substitution and must be tested (e.g., alternating, \(p\)-series, comparison).

About the Radius of Convergence Calculator

This calculator accepts coefficients \(\{a_n\}\) and an optional center \(x_0\) (default \(0\)). It computes \(R\) via the ratio or root test, using $$R=\left(\limsup_{n\to\infty}\sqrt[n]{|a_n|}\right)^{-1}\quad\text{or}\quad R=\left(\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|\right)^{-1}.$$ It then reports the open interval of convergence \(|x-x_0|<R\) and prompts you to test the boundary series at \(x=x_0\pm R\). For clarity and accessibility, formulas can be displayed with MathJax (responsive, mobile-friendly), while numeric limits and simplifications can be computed using .

How to Use this Radius of Convergence Calculator

  1. Enter the coefficient rule \(a_n\) and the center \(x_0\).
  2. Select a method: root test \(R=1/\limsup \sqrt[n]{|a_n|}\) or ratio test \(R=1/\lim |a_{n+1}/a_n|\).
  3. Compute \(R\), view the interval \(|x-x_0|<R\), and then evaluate both endpoints separately.
  4. Record the final interval of convergence (open, half-closed, or closed) based on endpoint tests.

Examples (using the same formulas)

Example 1: \(\displaystyle \sum_{n=0}^{\infty}\frac{3^n}{n!}(x-1)^n\).
Ratio: \(\left|\frac{a_{n+1}}{a_n}\right|=\frac{3}{n+1}\to 0\Rightarrow R=\frac{1}{0}=\infty.\) Converges for all \(x\).

Example 2: \(\displaystyle \sum_{n=1}^{\infty} n\,(x+2)^n\) (center \(-2\)).
Root: \(\sqrt[n]{|a_n|}=\sqrt[n]{n}\to 1\Rightarrow R=\frac{1}{1}=1.\)
Endpoints: \(x=-3\Rightarrow \sum n(-1)^n\) diverges; \(x=-1\Rightarrow \sum n\) diverges. Interval of convergence: \((-3,-1)\).

Example 3: \(\displaystyle \sum_{n=0}^{\infty}\frac{(-1)^n n^2}{5^n}x^n\) (center \(0\)).
Root: \(\sqrt[n]{|a_n|}=\frac{\sqrt[n]{n^2}}{5}\to \frac{1}{5}\Rightarrow R=5.\)
Endpoints: \(x=\pm 5\Rightarrow \sum (\pm 1)^n n^2\) diverges. Interval: \((-5,5)\).

FAQs

Q1: Ratio or root test—when should I use each?
Use whichever is easier: ratios for factorials/powers, roots for general exponentials. Both give the same \(R\) when limits exist.

Q2: What does \(R=\infty\) mean?
The series converges for every real \(x\) (entire radius), though complex-plane behavior may also be considered in advanced contexts.

Q3: Do I always need to test endpoints?
Yes. The formula gives \(|x-x_0|<R\); endpoints require separate convergence tests (e.g., alternating test, comparison, \(p\)-series).

Q4: What if the ratio limit doesn’t exist?
Use the root test with \(\limsup\). If still ambiguous, analyze subsequences or apply comparison/alternating tests at endpoints.

Q5: Can the calculator handle zero coefficients?
Yes. Zeros are allowed; limits are taken over the full sequence \(|a_n|\). Sparse patterns may still be analyzed via \(\limsup\).