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

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

Step-by-step derivation

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

|a_n|=3·(8/5)^n

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

absolute ratio=8/5

r=8/5>1

|a_n|≥A=3>0 for every n≥0

convergent series ⇒ its terms tend to0

these terms do not tend to0 ⇒ series diverges

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.