Calculator Form
Formula used
Ratio test formula
For a power series Σaₙ(x-c)ⁿ, if L = lim |aₙ₊₁ / aₙ| exists, then the radius is R = 1 / L.
Root test formula
If C = limsup |aₙ|^(1/n), then the radius is R = 1 / C. This works well for nth powers or exponential-style coefficients.
Convergence rule
The series converges absolutely when |x-c| < R and diverges when |x-c| > R. Endpoints where |x-c| = R must be checked separately.
How to use this calculator
- Choose the ratio test, root test, or direct radius option.
- Enter the series center c from the expression Σaₙ(x-c)ⁿ.
- Type the known limit value or radius, depending on the selected method.
- Set the endpoint behavior if you already know whether each boundary converges or diverges.
- Add a sample x value to classify one point against the radius.
- Press Calculate Radius to show the result above the form, view the graph, and export CSV or PDF reports.
Example data table
| Power series | Test idea | Center c | Radius R | Interval note |
|---|---|---|---|---|
| Σxⁿ | Ratio limit L = 1 | 0 | 1 | (-1, 1), then test endpoints |
| Σ[(x-2)ⁿ / 3ⁿ] | Root limit C = 1/3 | 2 | 3 | (-1, 5), then test endpoints |
| Σ[n!(x+1)ⁿ] | Ratio limit grows without bound | -1 | 0 | Converges only at x = -1 |
| Σ[(x-4)ⁿ / n²] | Root limit C = 1 | 4 | 1 | (3, 5), endpoints need separate tests |
FAQs
1) find the radius of convergence of the power series
Write the series as Σaₙ(x-c)ⁿ. Use L = lim |aₙ₊₁/aₙ| or C = limsup |aₙ|^(1/n). Then compute R = 1/L or R = 1/C. After that, test both endpoints separately.
2) How to find the radius of convergence of a power series
Identify the coefficients aₙ and the center c. Choose the ratio or root test, find the needed limit, invert it to get R, then build the interval c-R to c+R and check endpoints.
3) What happens if the ratio or root limit equals zero?
If the limit is 0, then R = ∞. The power series converges for every real x, so the interval of convergence becomes all real numbers.
4) What happens if the ratio or root limit becomes infinite?
If the limit is ∞, then R = 0. The series converges only at the center x = c, not on any open interval around that center.
5) Why must endpoints be tested separately?
At |x-c| = R, the ratio and root tests do not decide convergence. One endpoint can converge while the other diverges, so each boundary needs its own test.
6) Can a power series have a center other than zero?
Yes. Many power series are centered at c instead of 0. The radius still measures the same distance from c to each boundary point.
7) Which test should I choose?
Use the ratio test when coefficients contain factorials or consecutive-term patterns. Use the root test when coefficients involve nth powers, exponentials, or limsup behavior.
8) Does every point inside the radius converge absolutely?
Yes. Every point with |x-c| < R converges absolutely for a power series. Outside the radius it diverges, and only boundary points need extra checking.