# Question 17965: Sum Absolute Terms, Not the Absolute Value of the Signed Sum

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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.

R0034-17965 · q-20261014-advance-20261104-r03-absolute-convergence-of-series-0015



## Original question and givens

Use terms a*r^k for k=0,1,... with exact a=15/16, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, certificate).


- G001 — Use terms a*r^k for k=0,1,... with exact a=15/16, r=-1/2 and |r|<1. Give (a, r, absolute_series_sum, absolutely_convergent, 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-absolute-convergence-of-series](https://ocme.evemisslab.com/en/mko/mko-absolute-convergence-of-series/)





## Step-by-step derivation

S001 · impliesBefore

source operation=absseries-geometric;original question snapshot retained;source presentation interface resolved in this page

After

EXPLICIT ordered output fields=[a, r, absolute_series_sum, absolutely_convergent, certificate]

Conditions: none

Preserve the original snapshot's historical interface marker while listing the full ordered fields here,without browser JavaScript or an external lookup.Field labels are presentation metadata,not mathematical evaluation.The fourth term-zero No marks insufficiency,not a mislabeled second convergence flag.

Step identifiers and dependenciesRule: absseries_explicit_source_interface_fields_without_javascript · Uses: G001





S002 · calculationBefore

absolute geometric series one minus effective ratio := sub(1, 1/2)

After

absolute geometric series one minus effective ratio = 1/2

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

absolute geometric series exact infinite candidate sum := div(15/16, 1/2)

After

absolute geometric series exact infinite candidate sum = 15/8

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

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





S004 · impliesBefore

absolute geometric series:original SIGNED a=15/16;SIGNED r=-1/2;absolute ratio=1/2<1;effective terms=(15/16)*(1/2)^k,k=0,1,...

After

candidate sum=15/8;effective denominator1-ratio=1/2 is nonzero

Conditions: none

Retain signed a and r.Change to effective absolute values only when this operation asks for the absolute series.The |r|<1 guard is this certificate's domain;finite samples or an unrelated zero-coefficient exception do not change it.

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





S005 · impliesBefore

absolute geometric series:for ANY N>=1,the first N terms start at k0 with leading=15/16,ratio=1/2

After

(1-ratio)*S_N=(15/16)*(1-(1/2)^N);S_N=(15/16)*(1-(1/2)^N)/(1/2);candidate minus S_N=(15/16)*(1/2)^N/(1/2)

Conditions: none

Multiply the finite sum by1−ratio and cancel adjacent powers,leaving endpoints.This identity holds for arbitrary N≥1,not a formula guessed from several partial sums.

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





S006 · impliesBefore

absolute geometric series:|effective ratio|=1/2<1;remainder magnitude=abs(15/16)*abs(1/2)^N/abs(1/2)

After

geometric power tends to0;remainder tends to0;partial sums converge to the exact candidate

Conditions: none

Powers of a ratio with absolute value strictly below1 tend to0.Fixed finite coefficients and a nonzero denominator preserve this remainder limit,which proves convergence of the partial-sum sequence.

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





S007 · impliesBefore

original signed a=15/16;r=-1/2;for EVERY k>=0,abs(a*r^k)=abs(a)*abs(r)^k;absolute-series sum=15/8

After

for every m>n>=1,absolute signed tail<=absolute-series tail=abs(a)*abs(r)^n/(1-abs(r)) tending to0;Cauchy criterion and real completeness give convergence of original series

Conditions: none

Absolute values act termwise,giving |a|/(1−|r|),not |a/(1−r)|.Every signed tail is bounded by an absolute tail tending to0;the Cauchy criterion and real completeness then give original-series convergence,not extrapolation from a finite prefix.

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









## Reverse checks


- S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Recompute absolute geometric series exact infinite candidate sum with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute absolute geometric series one minus effective ratio with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(15/16, -1/2, 15/8, Yes, absolute_geometric_certificate)

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: 24f5d7e1dbaff7c44abdc9e2d8f8024ef61c96df0a18d634d516404c05fb6375
- candidate record SHA-256: 2ded31f126f0cf88bf2f7620dd478915b21305b13699f3e69342a23607d41a05
- semantic review receipt SHA-256: f73d099ebe3ff1c69a5d96f08c75b11b9e5b05120af40cf3ef388fce3b5c5336
- question-first witness SHA-256: 22675ffb6209b4fea076032b29b96e3dc19aad4244c54493f5f65f5e5d875699
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261104-r03-absolute-convergence-of-series-0015.json)