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Question 26978: A Convergent Nonnegative Geometric Upper Series Controls the Smaller Series

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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← Question derivation index · Original question batch

R0052-26978 · q-20261015-advance-20261117-r01-series-comparison-test-0003

Original question and givens

Use exact scales A=3,B=4, rational ratio r=3/9. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

  • G001 — Use exact scales A=3,B=4, rational ratio r=3/9. For n≥0 use a_n=A*(r)^n,b_n=B*(r)^n, with0≤A≤B and0≤r<1. Give (a_scale, b_scale, ratio, sum_a, sum_b, sum_comparison, certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source roles = [series-upper-geometric]

After

derived = [[a_scale, b_scale, ratio, sum_a, sum_b, sums_ordered, certificate]]

Conditions: none

List this operation/mode exact source field order without JavaScript.Keep Boolean,string and null roles distinct;do not pad fields or infer the interface from answer shape.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [3, 4, 1/3]

After

derived = [EVERY tail term is A*r^k and B*r^k for k>=0, r^0=1 including r=0, Yes]

Conditions: none

The whole tail law is given,with k starting at0.Nonnegative A<=B and r give termwise comparison;0<=r<1 licenses the geometric sum.At r=0 the first term still uses r^0=1.

Step identifiers and dependencies

Rule: Series_WHOLE_geometric_tail_premises_not_finite_samples · Uses: G001, S001

S003 · calculation

Before

Series ORIGINAL one minus ratio := sub(1, 1/3)

After

Series ORIGINAL one minus ratio = 2/3

Conditions: none

Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Series ORIGINAL geometric tailA := div(3, 2/3)

After

Series ORIGINAL geometric tailA = 9/2

Conditions: none

Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Series ORIGINAL geometric tailB := div(4, 2/3)

After

Series ORIGINAL geometric tailB = 6

Conditions: none

Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

source roles = [3, 4, 1/3, 2/3]

After

derived = [9/2, 6, Yes]

Conditions: none

For the finite sum,multiplication by(1-r) telescopes to A(1-r^N);r^N tends to0,hence the tail sum A/(1-r),and similarly for B.The nonnegative a partial sums increase and are bounded by the convergent b sum.This is not a finite-sample replacement for all n.

Step identifiers and dependencies

Rule: Series_GENERAL_geometric_partial_sum_limit_and_upper_comparison · Uses: G001, S005

Reverse checks

  • S006 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
  • S005 · premise_and_result_audit — Recompute Series ORIGINAL geometric tailB with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Series ORIGINAL geometric tailA with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Series ORIGINAL one minus ratio with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
  • S001 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.

Result

(3, 4, 1/3, 9/2, 6, Yes, upper_comparison_proves_convergence)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

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: 3377ce8a78b0ed212da509e7254163943b1a9ef67e0bd036cfdea3d9f1a4acb0
  • candidate record SHA-256: a50a260530bd4832a42c2813f62004754525a9891edb8b8a860d8ca8a05b597e
  • semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252
  • question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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