# Question 26939: Arbitrary Finite Prefixes Do Not Change Convergence of Dominated Tails

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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.

R0052-26939 · q-20261015-advance-20261117-r01-series-comparison-test-0064



## Original question and givens

Use exact scales A=14,B=15, rational ratio r=4/11. Prefix a=[14, 15, 16], prefix b=[1, 2, 3], each with cutoff=3 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=14,B=15, rational ratio r=4/11. Prefix a=[14, 15, 16], prefix b=[1, 2, 3], each with cutoff=3 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.


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





## Step-by-step derivation

S001 · impliesBefore

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 dependenciesRule: source_ordered_fields_without_JS · Uses: G001





S002 · impliesBefore

source roles = [14, 15, 4/11]

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 dependenciesRule: Series_WHOLE_geometric_tail_premises_not_finite_samples · Uses: G001, S001





S003 · calculationBefore

Series prefixA ORIGINAL running sum0 := sum(0, 14)

After

Series prefixA ORIGINAL running sum0 = 14

Conditions: none

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

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





S004 · calculationBefore

Series prefixA ORIGINAL running sum1 := sum(14, 15)

After

Series prefixA ORIGINAL running sum1 = 29

Conditions: none

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

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





S005 · calculationBefore

Series prefixA ORIGINAL running sum2 := sum(29, 16)

After

Series prefixA ORIGINAL running sum2 = 45

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

Series prefixB ORIGINAL running sum0 := sum(0, 1)

After

Series prefixB ORIGINAL running sum0 = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · calculationBefore

Series prefixB ORIGINAL running sum1 := sum(1, 2)

After

Series prefixB ORIGINAL running sum1 = 3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · calculationBefore

Series prefixB ORIGINAL running sum2 := sum(3, 3)

After

Series prefixB ORIGINAL running sum2 = 6

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · impliesBefore

source roles = [3, [14, 15, 16], [1, 2, 3]]

After

derived = [45, 6, 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 dependenciesRule: Series_arbitrary_nonnegative_prefix_count_NOT_global_order · Uses: G001, S008





S010 · calculationBefore

Series ORIGINAL one minus ratio := sub(1, 4/11)

After

Series ORIGINAL one minus ratio = 7/11

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009





S011 · calculationBefore

Series ORIGINAL geometric tailA := div(14, 7/11)

After

Series ORIGINAL geometric tailA = 22

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · calculationBefore

Series ORIGINAL geometric tailB := div(15, 7/11)

After

Series ORIGINAL geometric tailB = 165/7

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · impliesBefore

source roles = [14, 15, 4/11, 7/11]

After

derived = [22, 165/7, 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 dependenciesRule: Series_GENERAL_geometric_partial_sum_limit_and_upper_comparison · Uses: G001, S012





S014 · calculationBefore

Series ORIGINAL totalA prefix plus tail := sum(45, 22)

After

Series ORIGINAL totalA prefix plus tail = 67

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013





S015 · impliesBefore

source roles = [3, 45, 6, 22, 165/7]

After

derived = [67, 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 dependenciesRule: Series_finite_prefix_preserves_tail_convergence_NOT_total_order · Uses: G001, S014









## Reverse checks


- S015 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
- S014 · premise_and_result_audit — Recompute Series ORIGINAL totalA prefix plus tail with exact rational arithmetic, not rounded decimals.
- S013 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
- S012 · premise_and_result_audit — Recompute Series ORIGINAL geometric tailB with exact rational arithmetic, not rounded decimals.
- S011 · premise_and_result_audit — Recompute Series ORIGINAL geometric tailA with exact rational arithmetic, not rounded decimals.
- S010 · premise_and_result_audit — Recompute Series ORIGINAL one minus ratio with exact rational arithmetic, not rounded decimals.
- 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 prefixB ORIGINAL running sum2 with exact rational arithmetic, not rounded decimals.
- S007 · premise_and_result_audit — Recompute Series prefixB ORIGINAL running sum1 with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Recompute Series prefixB ORIGINAL running sum0 with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Recompute Series prefixA ORIGINAL running sum2 with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute Series prefixA ORIGINAL running sum1 with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute Series prefixA ORIGINAL running sum0 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, 45, 6, 22, 165/7, 67, 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: 6875d6df65d6e911299e1a8a1a26c021465fc9cc27f6ac4fd687bcba6cea4675
- candidate record SHA-256: 414cee0d71e5655c2e4462e65117f8411eeb10ef925912364946fb96f4ca377c
- semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252
- question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2
- Lean: NOT_RUN · human review: NOT_PERFORMED

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