Radius of convergence of a power series

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

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Infinite series — Convergence is checked at each point.
• Supremum and infimum — The convergence set gives an extremal radius description.
• Absolute value — Use distance from the center and absolute convergence.
• Natural-number powers and principal square roots — 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.