Radius of Convergence Calculator
Use the ratio test on a power series and see every step.
Enter the coefficients a0 to a7 of your power series, or load one of the worked examples below.
Radius of convergence (R)
Interval of convergence:
Ratio test steps
| Step | a_n | a_(n+1) | |a_n / a_(n+1)| |
|---|
How the ratio test works
A power series is a sum of terms a_n times x^n. The ratio test divides each coefficient by the next one. When those ratios settle on one number, that number is the radius of convergence. The series converges for x values closer to the centre than R, and diverges for x values further away than R.
Worked examples
- x^n / 2^n: coefficients 1, 0.5, 0.25, 0.125. Every ratio equals 2, so R = 2. The series converges when |x| < 2.
- x^n / n!: coefficients 1, 1, 0.5, 0.167. The ratios are 1, 2, 3, 4 and keep growing, so R is infinite. The series converges for every x.
- n! x^n: coefficients 1, 1, 2, 6, 24. The ratios are 1, 0.5, 0.33, 0.25 and keep shrinking toward 0, so R = 0. The series converges only at x = 0.
The endpoints of the interval, x = R and x = -R, need a separate check. The ratio test alone does not settle them.
Frequently Asked Questions
Related Tools
Check Out Other Popular Tools
GIF to JXL Converter — Free Online GIF to JPEG XL Converter
Convert GIF images to JPEG XL (.jxl) online for free. Private in-browser conversion — no uploads, no sign-up, no watermark.
Dice Roller & Coin Flipper
Roll 3D polyhedral dice (d4, d6, d8, d10, d12, d20, d100) or flip coins with 100% cryptographic fairness, D&D advantage, and sound effects.
Hash Generator
Generate MD5, SHA1, SHA256, SHA512 hashes from text or files. Free online hash generator with instant results. Secure client-side processing.
Was this tool helpful?
Comments
Loading comments...