Radius of Convergence Calculator

Enter coefficients \(a_n\) for the power series \(\sum a_n (z-z_0)^n\). Use commas, spaces or new lines. For formulas, enter a simple expression with n, e.g. 1/n! or n^2.

Estimates shown use numerical approximations (root test and ratio test). For rigorous proofs, consult analytic texts.

How to use the Convergence Radius Calculator

A convergence radius calculator computes the radius of convergence RR of a complex power series ∑n=0∞an(z−z0)n∑n=0∞​an​(z−z0​)n numerically, typically using the root test (R=1/lim sup⁡n→∞∣an∣1/nR=1/limsupn→∞​∣an​∣1/n) and/or the ratio test (R=lim⁡n→∞∣an/an+1∣R=limn→∞​∣an​/an+1​∣) to estimate where the series converges.

What this Convergence Radius Calculator does

This tool numerically estimates the radius of convergence for a power series from the coefficients anan​. It accepts a list of explicit coefficients or a formula in terms of nn (for example 1/factorial(n) or 1/(n^2)). It computes the sequence ∣an∣1/n∣an​∣1/n (used in the root test) and the sequence ∣an/an+1∣∣an​/an+1​∣ (used in the ratio test), provides numerical limsup/liminf estimates of these sequences, and converts them to radius estimates RR. A Plotly.js chart visualizes ∣an∣1/n∣an​∣1/n vs nn so you can see tail behavior and judge convergence numerically.

Why this tool is useful

Users typically look for quick, reliable methods to tell whether a series converges and, if so, how far from the center z0z0​that holds. This calculator matches that need by:

  • letting you paste coefficients directly,
  • handling common formulas like factorials or powers,
  • showing both root- and ratio-based estimates, and
  • plotting the sequence so you can visually inspect the limiting behavior.

The calculator is intended for exploratory or instructional use and gives strong numeric intuition for series like exponentials, binomial series, and rational sequences.

How to use the calculator (step-by-step)

1. Provide coefficients or a formula

  • If you have explicit coefficients, paste them separated by commas or spaces: 1, 1/2, 1/6, 1/24, ...
  • If the coefficients follow a rule, enter a formula using n. Examples:
    • 1/factorial(n) for an=1/n!an​=1/n!
    • 1/(n^2) for an=1/n2an​=1/n2
    • n^3 for an=n3an​=n3

The tool supports factorial(n) and common Math functions.

2. Adjust number of terms and start index

  • Number of terms controls how many anan​ values to compute and plot. For smooth tail behavior, use a reasonably large value (100–500). The default is 200 for a good balance of speed and accuracy.
  • Start index allows you to offset if your sequence begins at n>0n>0.

3. Compute and read results

  • Click Compute. The results area reports:
    • numerical limsup |a_n|^{1/n} and the root-test estimate R=1/lim sup⁡R=1/limsup,
    • numerical tail liminf for ∣an/an+1∣∣an​/an+1​∣ and the ratio-test estimate,
    • a Plotly chart showing ∣an∣1/n∣an​∣1/n vs nn and the computed limsup line.
  • If the root limsup is 0, the tool reports R=∞R=∞ (entire function). If values diverge or are unstable, the plot helps you see irregular behavior.

Interpreting the output

  • If both root and ratio estimates agree, you have a reliable numeric estimate of RR.
  • If estimates disagree or values are noisy, increase the number of terms or inspect the plotted tail. Oscillatory coefficients may require analytic methods.
  • Use the plot to check if ∣an∣1/n∣an​∣1/n stabilizes; if it approaches a horizontal limit, the root test estimate is trustworthy.

Practical tips & common examples

  • 1/factorial(n) → root limsup ≈ 0, so R=∞R=∞. Good for ezez.
  • 1 (all ones) → ∣an∣1/n=1∣an​∣1/n=1 → R=1R=1.
  • n! → root limsup grows large → R=0R=0 (no positive radius).
  • For rational sequences, the ratio test often gives a clean limit.

Technical notes on numeric estimation

The tool uses pragmatic numeric heuristics: it computes the sequence for many terms and takes the maximum of a tail fraction to approximate limsup, and the minimum of a tail fraction for liminf. These numerical proxies are practical and effective for most well-behaved sequences, but they are not a formal proof. Use rigorous analytic tests for formal statements.

Accessibility and embedding

  • The calculator is designed to be dropped into a WordPress Custom HTML block and sized to fit a typical content column (max 740px). It is responsive and will shrink to smaller widths.
  • If your WordPress strips <script> tags, add the JavaScript to your theme or child theme and keep the HTML wrapper in the post.

Disclaimer

This calculator provides numerical estimates and visual intuition only. It is not a substitute for rigorous mathematical proof. Numerical approximations may mislead for oscillatory or pathological coefficient sequences. Use symbolic or analytic methods when a formal proof is required.

FAQ

Q: Which test should I trust — root or ratio?
A: Use the root test for factorial or exponential-type coefficients; ratio is often best for rational or recurrence-defined coefficients. If both agree, trust the common value.

Q: What if the plot looks noisy?
A: Increase the number of terms, check your coefficient input for errors, or derive analytic asymptotics.

Q: Can the tool handle alternating coefficients?
A: Yes — it computes absolute values for the root and ratio tests. Alternation does not affect radius, only boundary behavior.

Q: Is this precise for formal proofs?
A: No. This tool is numeric/heuristic. For formal results, use analytic techniques.

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