Calculator Form
Example Data Table
Example series: Σ ((x - 2)ⁿ / 3ⁿ). Here, aₙ = 1 / 3ⁿ, so the radius is 3.
| n | aₙ | |aₙ|^(1/n) | |aₙ / aₙ₊₁| |
|---|---|---|---|
| 1 | 0.3333333333 | 0.3333333333 | 3 |
| 2 | 0.1111111111 | 0.3333333333 | 3 |
| 3 | 0.0370370370 | 0.3333333333 | 3 |
| 4 | 0.0123456790 | 0.3333333333 | 3 |
| 5 | 0.0041152263 | 0.3333333333 | 3 |
| 6 | 0.0013717421 | 0.3333333333 | 3 |
Formula Used
For a power series of the form Σ aₙ(x - c)ⁿ, the radius of convergence is determined from the coefficients.
- Ratio test: If the limit exists, then R = lim |aₙ / aₙ₊₁|.
- Root test: If the limit exists, then R = 1 / lim |aₙ|^(1/n).
- Cauchy-Hadamard formula: 1 / R = limsup |aₙ|^(1/n).
- Interval form: The series converges whenever |x - c| < R.
- Endpoints: Values at x = c - R and x = c + R require separate testing.
When only a finite list of coefficients is available, this page estimates the limsup using the last few root values from the list.
How to Use This Calculator
- Choose a method that matches the information you already have.
- Enter the center c of the power series.
- Provide either local coefficients, a limsup value, or a coefficient list.
- Optionally mark endpoint behavior if you already tested those values separately.
- Press Calculate Radius to show the result above the form.
- Review the summary table, graph, and diagnostic table.
- Use the export buttons to download the current result as CSV or PDF.
Frequently Asked Questions
1) What does the radius of convergence tell me?
It gives the distance from the center where the power series definitely converges. Inside that radius, convergence holds. Outside it, divergence holds. Endpoints must be checked separately.
2) Why can the endpoints behave differently?
A power series can converge at one endpoint, both, or neither. The radius alone does not decide endpoint behavior, so each endpoint needs an independent convergence test.
3) When should I use the ratio test option?
Use it when you know consecutive coefficients and the ratio limit is expected to stabilize. It is common for factorial, exponential, and geometric-style coefficient patterns.
4) When is the root test more useful?
The root test is helpful when coefficients involve powers or when nth-root behavior is easier to estimate than consecutive ratios. It aligns directly with the Cauchy-Hadamard formula.
5) Why is the coefficient-list method only an estimate?
A true radius depends on infinitely many coefficients. A finite list can only approximate the limiting behavior, so the displayed radius is a practical estimate, not an exact proof.
6) What does an infinite radius mean?
It means the series appears to converge for every real x. Entire functions, like eˣ and sine, have power series with infinite radius of convergence.
7) Can zero coefficients affect the estimate?
Yes. Zero coefficients can make local ratios undefined or unusually large. That is why the coefficient-list mode also shows root values, which often remain more informative.
8) Does changing the center change the radius?
The center changes the interval location, but not the coefficient-based radius itself for the same power series representation. The radius still measures distance from the center.