# Ratio limits, geometric domination and L=1 [← Guide index](https://ocme.evemisslab.com/en/derivations/) · [Original MKO](https://ocme.evemisslab.com/en/mko/mko-ratio-test-series/) This is an owner-adopted natural-language teaching guide. Finite/arithmetic checks are not universal proofs; no Lean was run, and review scope is recorded per revision. Original source status labels are retained as historical data. Formulas retain exact TeX; no new semantic AST is invented. derivation-mko-ratio-test-series-01 · ocme-new142-reader-adoption-20261021 ## Goal and scope Use complete rational ratios for L<1 convergence and L>1 divergence, then audit indecision at L=1. Qualify eventual nonzero terms and absolute ratio limits, not limits inferred from finite ratios. ## Givens and symbols - algebra - epsilon - real_complete - G1 — a_n=(1/2)^n/(n+1), b_n=2^n/(n+1) for every n≥0; also c_n=1/(n+1), d_n=1/(n+1)^2. - L — proved absolute-ratio limit - q — domination ratio between L and one ## Rules and sources Some rule statements have reader clarifications in this revision: they are paraphrases, not verbatim quotations. Referenced original source bytes and hashes are unchanged. - mko_definition · cited — Declare the existing limit L=lim|a_(n+1)/a_n| on an eventually nonzero tail. When L<1 choose L1 the term magnitudes eventually fail to tend to 0, 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. · mko-ratio-test-series - algebra · prerequisite — Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities. - epsilon · prerequisite — Convergence requires for every positive epsilon an integer N controlling every n≥N by a strict error bound; use the Archimedean property of the reals. - real_complete · prerequisite — Every nonempty bounded-above real set has a supremum, so bounded monotone real sequences converge, including bounded partial sums of nonnegative series. ## Main derivation S01 · equivalentBefore a_n>0\quad(n\ge0) After \left|a_{n+1}/a_n\right|=\tfrac12\frac{n+1}{n+2}=\tfrac12-\frac1{2(n+2)} Conditions: none Rule: algebra · Uses: G1 Leaf basis: algebra All terms are nonzero, and the full absolute-ratio identity directly supplies its limit. S02 · impliesBefore \frac1{2(n+2)}\to0 After L_a=1/21/(2epsilon) controls the error, proving the actual limit and an explicit intermediate q. S03 · impliesBefore a_{n+1}/a_n<1/2<3/4\quad(n\ge0) After 01,\quad b_{n+1}/b_n\ge3/2\ (n\ge2) Conditions: none Rule: algebra · Uses: G1 Leaf basis: algebra The full difference formula proves limit two, and for n≥2 the ratio is at least 3/2. S06 · impliesBefore b_n\ge b_2(3/2)^{n-2}\quad(n\ge2) After b_n\not\to0;\quad\sum b_n\text{ diverges} Conditions: none Rule: algebra · Uses: S05 Leaf basis: algebra Induction gives geometric growth and unbounded terms, violating the necessary term condition for series convergence. S07 · impliesBefore c_{n+1}/c_n=\frac{n+1}{n+2},\quad d_{n+1}/d_n=\left(\frac{n+1}{n+2}\right)^2 After L_c=L_d=1 Conditions: none Rule: algebra · Uses: G1 Leaf basis: algebra The errors of the two rational ratios relative to 1 tend to zero, giving the same limit 1; their denominators do not tend to zero, and neither ratio is constant. S08 · impliesBefore \sum c_n\text{ diverges by dyadic blocks};\quad\frac1{(n+1)^2}\le\frac1{n(n+1)}\ (n\ge1) After \sum d_n\text{ converges by telescoping upper comparison};\quad L=1\text{ is inconclusive} Conditions: none Rule: epsilon · Uses: G1, S07 Leaf basis: epsilon Each harmonic dyadic block is at least 1/2; reciprocal-square tails are bounded above by a telescoping series of sum one, giving opposite statuses at the same L. ## Substeps and lemmas No substeps in this guide. ## Reverse audits - S08 · premise_audit · G1, S07 — An inconclusive test does not mean the series has no convergence status. - S07 · premise_audit · G1 — A limit of one is not confused with every ratio equaling one. - S06 · premise_audit · S05 — A single ratio above one is not treated as a divergence proof. - S05 · premise_audit · G1 — The greater-than-one bound holds on the entire tail. - S04 · premise_audit · S03, real_complete — Bounded-monotone real convergence is assumed, with the upper bound independently supplied by a geometric sum. - S03 · premise_audit · S01, S02 — Domination covers every n, not merely one favorable ratio. - S02 · premise_audit · S01 — All comparisons in L