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R0034-17763 · q-20261014-advance-20261103-r03-infinite-series-0063
Use c=13, positive shift s=2, and n=13 terms c/(s+k)−c/(s+k+1) for k=0 through n−1. Give (c,shift,n,partial_sum,remainder,infinite_sum,exact_telescoping_remainder). The source directly formats remainder=c/(s+n) and infinite_sum=c/s and may leave those two ratios unreduced; retain that exact text.
G001 — Use c=13, positive shift s=2, and n=13 terms c/(s+k)−c/(s+k+1) for k=0 through n−1. Give (c,shift,n,partial_sum,remainder,infinite_sum,exact_telescoping_remainder). The source directly formats remainder=c/(s+n) and infinite_sum=c/s and may leave those two ratios unreduced; retain that exact text.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.
Before
telescoping final remainder denominator := sum(2, 13)
After
telescoping final remainder denominator = 15
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Rule: exact_rational_arithmetic · Uses: G001
Before
telescoping first endpoint numeric value := div(13, 2)
After
telescoping first endpoint numeric value = 13/2
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
telescoping signed remainder numeric value := div(13, 15)
After
telescoping signed remainder numeric value = 13/15
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
telescoping reduced partial sum := sub(13/2, 13/15)
After
telescoping reduced partial sum = 169/30
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
telescoping partial plus signed remainder := sum(169/30, 13/15)
After
telescoping partial plus signed remainder = 13/2
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
original c=13;shift=2;n=13;terms c/(shift+k)-c/(shift+k+1),k0..n-1
After
ALL interior terms cancel;S_n=c/shift-c/(shift+n)=169/30;signed remainder=13/15;reverse sum=13/2
Conditions: none
Every interior positive endpoint cancels its preceding negative occurrence,leaving the first endpoint and final negative one.Add the signed remainder back to reverse-check;do not replace a negative-c remainder by its absolute value.
Rule: series_telescoping_all_interior_cancellation_and_reverse_check · Uses: G001, S005
Before
same source formatting c=13;source remainder ratio text=13/15;source infinite-sum ratio text=13/2
After
computed partial sum REDUCED=169/30;remainder/infinite fields retain DIRECT UNREDUCED source text;numeric equivalents=[13/15, 13/2]
Conditions: none
The source reduces the computed partial sum but directly formats the other two ratio fields,which can remain2/4 or0/den.Preserve those answer bytes and verify their numerical values separately;mathematical equivalence does not authorize rewriting the source output.
Rule: series_telescoping_source_ratio_bytes_not_silent_reduction · Uses: G001, S006
Before
DECLARED whole telescoping tail;remainder at general n is(13)/(n+2)
After
signed remainder tends to0;full partial-sum limit=c/shift under the given index convention
Conditions: none
The fixed coefficient and positive shift give an explicit remainder for every n.Its limit proves the full sum,not success of the single supplied endpoint check.
Rule: series_telescoping_full_remainder_limit_not_one_n · Uses: G001, S007
S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 — Recompute telescoping partial plus signed remainder with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute telescoping reduced partial sum with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute telescoping signed remainder numeric value with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute telescoping first endpoint numeric value with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Recompute telescoping final remainder denominator with exact rational arithmetic, not rounded decimals.(13, 2, 13, 169/30, 13/15, 13/2, exact_telescoping_remainder)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: d327215aec9281a684b8ca4c64f80c3422255533d99fd594f0036a6fb156d9f2candidate record SHA-256: ad2da0d718c87bd9411a55bcc9fc3580d89630fe92d769f9f702d22c2e7420bcsemantic review receipt SHA-256: f73d099ebe3ff1c69a5d96f08c75b11b9e5b05120af40cf3ef388fce3b5c5336question-first witness SHA-256: 22675ffb6209b4fea076032b29b96e3dc19aad4244c54493f5f65f5e5d875699Lean: NOT_RUN · human review: NOT_PERFORMED