# Radius of convergence of a power series

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Around its center c, a power series has absolute convergence inside radius R and divergence outside; endpoints require separate decisions.

mko-power-series-radius-of-convergence · definition · v0.10.0



## Mathematical statement

For sum over n≥0 of a_n*(x−c)^n, R may be0, finite positive or infinity. Distance less than R gives absolute convergence and distance greater than R gives divergence; equality cannot be decided from R alone. This slice declares geometric coefficients q^n, endpoint coefficients R^(−n)/(n+1)^p, and two complete laws sharing zeros at every index0≤n≤N before distinct scaled geometric tails. A ratio test needs its relevant limit and is not a general definition. A finite coefficient prefix cannot determine a general radius, and a bounded-real-only supremum rule must not exclude0 or infinity.





## Representative formula (exact TeX)

R(f,c)=r





## Assumptions


- Every exact family declares its coefficient law for all n: geometric q^n, endpoint R^(-n)/(n+1)^p, factorial extremes or complete geometric tails after a shared zero prefix.
- Radii and point positions are exact rationals or explicit0/infinity markers; endpoints are classified separately.





## Symbol scope


- R — radius of convergence
- f — power series
- c — series center
- r — extended nonnegative radius





## Intuition, definition, and misconceptions

For the whole geometric law a_n=q^n, R=1/|q|.

The endpoint law R^(−n)/(n+1)^p shows different convergence types at the two ends.

Radius0, radius infinity and complete laws sharing a finite zero prefix but having different tails are valid boundaries.

Do not include endpoints automatically, conclude no radius exists when a ratio limit fails or determine a unique radius from a finite prefix.





## Adopted derivation guide

[Radius, endpoints and complete coefficient laws](https://ocme.evemisslab.com/en/derivations/derivation-mko-power-series-radius-of-convergence-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Infinite series](https://ocme.evemisslab.com/en/mko/mko-infinite-series/) — Convergence is checked at each point.
- [Supremum and infimum](https://ocme.evemisslab.com/en/mko/mko-supremum-and-infimum/) — The convergence set gives an extremal radius description.
- [Absolute value](https://ocme.evemisslab.com/en/mko/mko-absolute-value/) — Use distance from the center and absolute convergence.
- [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/mko/mko-power-root/) — Coefficient growth and root behavior matter.





## Verification boundary

Exact certificates concern declared full coefficient laws; general series need limsup or another complete argument.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add the real exponential function and analytic ordinary generating functions.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/power-series-radius-of-convergence.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-power-series-radius-of-convergence/index.md)