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Question 26946: Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

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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R0052-26946 · q-20261015-advance-20261117-r01-series-comparison-test-0071

Original question and givens

Use exact scales A=21,B=22, rational ratio r=1/8. Prefix a=[], prefix b=[], each with cutoff=0 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).

  • G001 — Use exact scales A=21,B=22, rational ratio r=1/8. Prefix a=[], prefix b=[], each with cutoff=0 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, 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-eventual-tail]

After

derived = [[cutoff, prefix_a_sum, prefix_b_sum, tail_a_sum, tail_b_sum, total_a, convergent, 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 = [21, 22, 1/8]

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 · implies

Before

source roles = [0, [], []]

After

derived = [0, 0, tail exponent n-cutoff starts at0]

Conditions: none

cutoff is the NUMBER of prefix terms.Prefixes are separately nonnegative but need not be ordered termwise or by sum.The tail starts at exponent0 when n=cutoff;empty prefixes are allowed.

Step identifiers and dependencies

Rule: Series_arbitrary_nonnegative_prefix_count_NOT_global_order · Uses: G001, S002

S004 · calculation

Before

Series ORIGINAL one minus ratio := sub(1, 1/8)

After

Series ORIGINAL one minus ratio = 7/8

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 tailA := div(21, 7/8)

After

Series ORIGINAL geometric tailA = 24

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 · calculation

Before

Series ORIGINAL geometric tailB := div(22, 7/8)

After

Series ORIGINAL geometric tailB = 176/7

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, S005

S007 · implies

Before

source roles = [21, 22, 1/8, 7/8]

After

derived = [24, 176/7, 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, S006

S008 · calculation

Before

Series ORIGINAL totalA prefix plus tail := sum(0, 24)

After

Series ORIGINAL totalA prefix plus tail = 24

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, S007

S009 · implies

Before

source roles = [0, 0, 0, 24, 176/7]

After

derived = [24, Yes]

Conditions: none

For sufficiently large N,the full partial sum is a fixed finite prefix plus a tail partial sum.Adding a fixed finite number preserves the limit,hence A converges.Compare tails only,not full-series totals,and impose no source numeric gate on an unused totalB.

Step identifiers and dependencies

Rule: Series_finite_prefix_preserves_tail_convergence_NOT_total_order · Uses: G001, S008

Reverse checks

  • S009 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
  • S008 · premise_and_result_audit — Recompute Series ORIGINAL totalA prefix plus tail with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
  • S006 · premise_and_result_audit — Recompute Series ORIGINAL geometric tailB with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Series ORIGINAL geometric tailA with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Series ORIGINAL one minus ratio with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
  • 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

(0, 0, 0, 24, 176/7, 24, Yes, finite_prefix_does_not_change_tail_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: 7b32cc6e50baa4775af5260ba05e871506d0fd130c58bfb39e772785c9f0d5eb
  • candidate record SHA-256: a3fa6a2bc8a47cd29b95cf90c1788aed5f5ab703cf96bd1744dd200a3eb32c0d
  • semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252
  • question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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