What is radius of convergence?
Half the length of the open interval where a power series converges.
Radius of Convergence Calculator
Use geometric form Σ c rⁿ or enter the ratio limit L = lim |aₙ₊₁/aₙ| to get R.
Choose a series model
Geometric mode uses common ratio; ratio-test mode uses lim |aₙ₊₁/aₙ| = L.
About this calculator
How to use Radius of Convergence Calculator, plus the formulas and logic behind the results.
Tool description
The radius R describes the largest distance from the center where a power series converges.
How to use
- Select geometric or ratio-test mode.
- Enter q (geometric) or the ratio coefficient L.
- Select Calculate to view R and the interval description.
- Check endpoints separately for full interval of convergence.
Formulas & reference
Geometric Σ (qx)ⁿ converges for |qx| < 1 → R = 1/|q|If lim |aₙ₊₁/aₙ| = L|x|, then R = 1/L (L>0)L = 0 → R = ∞; L = ∞ → R = 0
Series tips
- Interval of convergence is (x₀−R, x₀+R) plus possible endpoints
- Geometric series is the model case for R
- Ratio test fails when L=1; try root test or comparison
- Center defaults to 0 unless you set x₀ for display
Calculation logic
Applies geometric convergence |q|<1 in x, or the ratio-test formula R = 1/L for lim |aₙ₊₁/aₙ| = L|x|.
Endpoint convergence is not decided here—only the radius R.
Radius of Convergence Calculator FAQ
Common questions about this calculator and how to use the results.
Geometric mode meaning?
For Σ c (q x)ⁿ, converge when |q x| < 1 so R = 1/|q|.
What is L in ratio mode?
The coefficient in lim |aₙ₊₁/aₙ| = L|x| for series centered at 0.
What if L = 0?
Radius is infinite (converges for all x).
Are endpoints included?
This tool reports R only; test x = ±R separately.
Is this free?
Yes.