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Question 708: An Absolute Geometric Ratio above One Violates the Necessary Term Limit

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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P1000-0708 · q-20261015-advance-20261117-r02-ratio-test-series-0033

Original question and givens

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

  • G001 — Use positive scale=8, exact ratio=8/5>1 and alternating=true. 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.

Step-by-step derivation

S001 · implies

Before

A=8>0; r=8/5>1; alternating=Yes

After

|a_n|=8·(8/5)^n

Conditions: none

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

Step identifiers and dependencies

Rule: absolute_geometric_term · Uses: G001

S002 · implies

Before

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

Rule: cancel_nonzero_factors · Uses: G001, S001

S003 · implies

Before

r=8/5>1

After

|a_n|≥A=8>0 for every n≥0

Conditions: none

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

Step identifiers and dependencies

Rule: nonzero_term_obstruction · Uses: G001, S002

S004 · implies

Before

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 dependencies

Rule: 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

(8, 8/5, Yes, 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: 7a30217db2946af51c6b0b79a15ccdf10ec7fd440685a7cb116ba3f4a10d7331
  • candidate record SHA-256: 2b47a6c5019c48995bd2fcea8c77e9c1f57f790ae0e0828e238579a9b4754af7
  • semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772
  • question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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