# Question 17763: Retain Both the Telescoping Remainder and the Source Ratio Text [Return to catalogue page 89](https://ocme.evemisslab.com/en/questions/page/89/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261103-r03-infinite-series-foundations/#q-20261014-advance-20261103-r03-infinite-series-0063) 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. R0034-17763 · q-20261014-advance-20261103-r03-infinite-series-0063 ## Original question and givens 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. ## 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-infinite-series](https://ocme.evemisslab.com/en/mko/mko-infinite-series/) ## Step-by-step derivation S001 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore 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. Step identifiers and dependenciesRule: series_telescoping_all_interior_cancellation_and_reverse_check · Uses: G001, S005 S007 · impliesBefore 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. Step identifiers and dependenciesRule: series_telescoping_source_ratio_bytes_not_silent_reduction · Uses: G001, S006 S008 · impliesBefore 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. Step identifiers and dependenciesRule: series_telescoping_full_remainder_limit_not_one_n · Uses: G001, S007 ## Reverse checks - 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. ## Result (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. ## 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: d327215aec9281a684b8ca4c64f80c3422255533d99fd594f0036a6fb156d9f2 - candidate record SHA-256: ad2da0d718c87bd9411a55bcc9fc3580d89630fe92d769f9f702d22c2e7420bc - 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-20261103-r03-infinite-series-0063.json)