# Question 703: An Absolute Geometric Ratio above One Violates the Necessary Term Limit

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

P1000-0703 · q-20261015-advance-20261117-r02-ratio-test-series-0028



## Original question and givens

Use positive scale=3, exact ratio=8/5>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).


- G001 — Use positive scale=3, exact ratio=8/5>1 and alternating=false. Give (scale, ratio, alternating, absolute_ratio, term_test_obstruction, conclusion).





## Definition and scope

Geometric terms with A>0 and |r|>1 satisfy |a_n|≥A; the terms do not tend to 0, so the series diverges.


- [mko-ratio-test-series](https://ocme.evemisslab.com/en/mko/mko-ratio-test-series/)





## Step-by-step derivation

S001 · impliesBefore

A=3>0; r=8/5>1; alternating=No

After

|a_n|=3·(8/5)^n

Conditions: none

Any alternating sign disappears in absolute value, and the entire base is exponentiated.

Step identifiers and dependenciesRule: absolute_geometric_term · Uses: G001





S002 · impliesBefore

|a_(n+1)|/|a_n|=(3·(8/5)^(n+1))/(3·(8/5)^n)

After

absolute ratio=8/5

Conditions: none

A and r are positive, so the complete common nonzero factor may be cancelled.

Step identifiers and dependenciesRule: cancel_nonzero_factors · Uses: G001, S001





S003 · impliesBefore

r=8/5>1

After

|a_n|≥A=3>0 for every n≥0

Conditions: none

With epsilon=A/2, no tail can make every term magnitude smaller than epsilon.

Step identifiers and dependenciesRule: nonzero_term_obstruction · Uses: G001, S002





S004 · impliesBefore

convergent series ⇒ its terms tend to0

After

these terms do not tend to0 ⇒ series diverges

Conditions: none

If partial sums converged, their successive differences a_n would tend to0; this necessary condition fails.

Step identifiers and dependenciesRule: necessary_term_limit · Uses: G001, S003









## Reverse checks


- S004 · premise_and_result_audit — Finite checks do not prove an infinite tail; retain this step's quantifiers, nonzero denominators and absolute values.
- S003 · premise_and_result_audit — Finite checks do not prove an infinite tail; retain this step's quantifiers, nonzero denominators and absolute values.
- S002 · premise_and_result_audit — Finite checks do not prove an infinite tail; retain this step's quantifiers, nonzero denominators and absolute values.
- S001 · premise_and_result_audit — Finite checks do not prove an infinite tail; retain this step's quantifiers, nonzero denominators and absolute values.





## Result

(3, 8/5, No, 8/5, terms_do_not_tend_to_zero, divergent)

The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 53569c5bf5d75d4b2a85e60836fd07f94fa24d0f503127768c764ccd6290b731
- candidate record SHA-256: 1c1357f653409845c6d3fdbbdb7d56850a7e0bf511dd86621adc040f809ed433
- semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772
- question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261117-r02-ratio-test-series-0028.json)