Ratio test for series

Mathematical statement

Declare the existing limit L=lim|a_(n+1)/a_n| on an eventually nonzero tail. When L<1 choose L<q<1 and establish eventual geometric domination to obtain absolute convergence. When L>1 the term magnitudes eventually fail to tend to0, so the series diverges. L=1 gives no decision: reciprocal-square and harmonic tails have opposite statuses with that same limit. Use absolute ratios and an actual limit argument, not a finite favorable prefix. Zero denominators and missing limits are separate boundary cases.

Representative formula (exact TeX)

R(a,L)=c

Assumptions

• Successive ratios are defined on some whole tail and their absolute-value limit exists.
• Comparison of L with1 uses exact rational values or a declared symbolic limit certificate.

Symbol scope

• R — ratio-test operator
• a — series terms
• L — absolute-ratio limit
• c — ratio-test conclusion

Intuition, definition, and misconceptions

Qualify eventual nonzero terms and the absolute-ratio limit first.

For L<1 retain L<q<1 and domination; separate L>1 from L=1.

Nonconstant reciprocal-power ratios give opposite L=1 witnesses, while finite prefixes and nonexistent limits do not qualify.

L=1 does not imply convergence, and a few signed ratios do not establish the needed absolute limit.

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

• Absolute convergence of series — Use the L<1 conclusion and series semantics.
• Sequence limits and exact tail cutoffs — Require the ratio limit.
• Absolute value — Use absolute rather than signed ratios.
• Comparison tests for series — Obtain geometric domination after choosing q.

Verification boundary

Geometric and reciprocal-power instances do not replace the general ratio-test proof. The current additive source revision writes the entire rational base in parentheses in A*(r)^n/(n+shift); original ambiguous displays remain in the retained historical source, not the active answers.

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 root test and limsup versions.

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.