radius of convergence calculator - Aaron Graves, PhDude Replica

Calculate the Radius of Convergence

Use one of the standard methods from power series analysis. You can compute exactly from a known limit, or estimate from a finite list of coefficients.

Center of series, c (for Σ a_n(x - c)ⁿ)

Method

Enter L = lim |aₙ / aₙ₊₁|

You may use decimals, fractions like 3/2, scientific notation, or "infinity".

Enter ρ = limsup |aₙ|^(1/n)

Then R = 1/ρ (with ρ = 0 giving R = ∞).

Enter coefficients a₀, a₁, ..., a_N

Separate by commas or spaces. Fractions like 1/6 are allowed.

This mode gives a numerical estimate of R from finite data.

What is the radius of convergence?

For a power series of the form Σ a_n(x - c)ⁿ, the radius of convergence, denoted by R, tells you how far from the center c the series converges absolutely.

If |x - c| < R, the series converges absolutely.
If |x - c| > R, the series diverges.
If |x - c| = R, each endpoint must be tested separately.

Core formulas used by this calculator

1) Ratio Test Formula

If the limit exists, R = lim_n→∞ |a_n/a_n+1|. This is often the fastest method when factorials or exponentials appear in coefficients.

2) Root Test Formula

In general, R = 1 / limsup_n→∞ |a_n|^1/n. This is very reliable and handles many cases where the ratio limit is awkward.

3) Finite-coefficient estimate

If you only have a finite list of coefficients, the calculator estimates the large-n behavior using tail values of both ratio-style and root-style expressions. This gives a practical approximation, but it is not a proof.

How to use this radius of convergence calculator

Enter the center c (default is 0).
Choose a method (Ratio, Root, or Coefficient Estimate).
Provide the needed value(s).
Click Calculate Radius.

The tool returns the radius R and the open interval (c - R, c + R) for real-variable interpretation.

Examples

Example A: Geometric-style coefficients

For Σ (x - 2)ⁿ / 3ⁿ, we have a_n = 1/3ⁿ. Ratio gives |a_n/a_n+1| = 3, so R = 3. Convergence is guaranteed for |x - 2| < 3.

Example B: Factorials in numerator

For Σ n! xⁿ, the ratio |a_n/a_n+1| = 1/(n+1) → 0, so R = 0. This means convergence only at x = 0.

Example C: Factorials in denominator

For Σ xⁿ/n!, |a_n/a_n+1| = n+1 → ∞, so R = ∞. The series converges for all real x.

Important endpoint reminder

Radius alone does not settle behavior at x = c ± R. You must test each endpoint separately (often with comparison, alternating-series, p-series, or integral tests depending on the resulting series).

Quick FAQ

Can radius be infinite?

Yes. If R = ∞, the power series converges for every real x.

Can radius be zero?

Yes. Then the series generally converges only at the center x = c.

Do I always need limsup?

The limsup form is the most general for root-test style analysis. If an ordinary limit exists, it is enough and easier to compute.

Summary

This calculator helps you compute or estimate the radius of convergence quickly using standard calculus tools. Use ratio or root methods for exact work when limits are known, and use coefficient estimation for exploratory numerical analysis.