Radius of Convergence Calculator
Use exact limits (ratio/root test) or estimate from a finite list of coefficients.
Ratio test (if limit exists): R = 1 / limn→∞ |an+1/an|
Root test (general): R = 1 / limsupn→∞ |an|1/n
A) Calculate from a known limit
B) Estimate from coefficient list
What is the radius of convergence?
When you have a power series like Σ an(x - c)n, the radius of convergence tells you how far from the center c you can move and still have the series converge.
If the radius is R, then:
- The series converges absolutely for |x - c| < R
- The series diverges for |x - c| > R
- The boundary points |x - c| = R require separate endpoint testing
Two core formulas
1) Ratio test formula
If the limit exists, L = lim |an+1/an|, then R = 1/L.
- L = 0 → R = ∞ (converges for all real x)
- L = ∞ → R = 0 (typically only at x = c)
2) Root test formula
In general, use S = limsup |an|1/n. Then R = 1/S.
The root formula is often the most robust because it still works even when the plain ratio limit fails to exist.
How to use this calculator
Exact mode
- Pick Ratio or Root method.
- Enter the corresponding limit value.
- Enter center c if your series is in powers of (x - c).
- Click Calculate Exact Radius.
Estimate mode
- Paste a finite list of coefficients (a0, a1, ...).
- Choose a numerical estimator (ratio or root).
- Click Estimate from Coefficients.
- Treat the result as approximate unless you have asymptotic justification.
Worked examples
Example 1: Geometric-style coefficients
Suppose an = 3n. Then |an+1/an| = 3, so L = 3 and R = 1/3.
Example 2: Factorial denominator
For an = 1/n!, the ratio is |an+1/an| = 1/(n+1) → 0. Therefore R = ∞.
Example 3: Harmonic coefficients
If an = 1/n, then |an|1/n → 1, so R = 1. Endpoint behavior then must be checked separately.
Interpreting results correctly
The radius gives a distance from c. On the real line, that suggests an open interval (c - R, c + R). But this does not automatically settle the endpoints. Always test x = c + R and x = c - R directly.
In complex analysis, think of a disk: |z - c| < R.
Common mistakes to avoid
- Forgetting absolute values in ratio/root formulas.
- Confusing c (center) with R (radius).
- Assuming endpoint convergence without testing.
- Treating a short numerical coefficient list as exact proof.
Quick FAQ
Can R be infinite?
Yes. Entire-function series (like ex) have R = ∞.
Can R be zero?
Yes. Some series only converge at the center x = c.
Do I always use ratio test?
Use ratio when convenient, but root/limsup is the general fallback. This calculator supports both.