# Question 24227: Exact Finite Exponential Sums Are Not General Infinite-Value Certificates [Return to catalogue page 122](https://ocme.evemisslab.com/en/questions/page/122/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261113-r02-real-exponential-function-foundations/#q-20261015-advance-20261113-r02-real-exponential-function-0002) 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. R0047-24227 · q-20261015-advance-20261113-r02-real-exponential-function-0002 ## Original question and givens Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=2/3, cutoff=6, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary). - G001 — Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=2/3, cutoff=6, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary). ## 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-real-exponential-function](https://ocme.evemisslab.com/en/mko/mko-real-exponential-function/) ## Step-by-step derivation S001 · impliesBefore source roles = [exp-finite-partial-sum] After derived = [[x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary]] Conditions: none List the source field order without JavaScript.Gradient boundary uses its original mode to choose6,8 or7 fields;no padding or answer-derived mode. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore source roles = [0] After derived = [1] Conditions: none The empty product0! is1,not0. Step identifiers and dependenciesRule: empty_factorial_is_multiplicative_unit · Uses: G001, S001 S003 · impliesBefore (2/3)^0 After Exp ORIGINAL x power0 = 1 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S002 S004 · calculationBefore Exp term0 power over factorial := div(1, 1) After Exp term0 power over factorial = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Exp cumulative sum through0 := sum(0, 1) After Exp cumulative sum through0 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Exp term1 factorial step1 := mul(1, 1) After Exp term1 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore (2/3)^1 After Exp ORIGINAL x power1 = 2/3 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S006 S008 · calculationBefore Exp term1 power over factorial := div(2/3, 1) After Exp term1 power over factorial = 2/3 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Exp cumulative sum through1 := sum(1, 2/3) After Exp cumulative sum through1 = 5/3 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Exp term2 factorial step1 := mul(1, 1) After Exp term2 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Exp term2 factorial step2 := mul(1, 2) After Exp term2 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore (2/3)^2 After Exp ORIGINAL x power2 = 4/9 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S011 S013 · calculationBefore Exp term2 power over factorial := div(4/9, 2) After Exp term2 power over factorial = 2/9 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Exp cumulative sum through2 := sum(5/3, 2/9) After Exp cumulative sum through2 = 17/9 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Exp term3 factorial step1 := mul(1, 1) After Exp term3 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Exp term3 factorial step2 := mul(1, 2) After Exp term3 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Exp term3 factorial step3 := mul(2, 3) After Exp term3 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · impliesBefore (2/3)^3 After Exp ORIGINAL x power3 = 8/27 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S017 S019 · calculationBefore Exp term3 power over factorial := div(8/27, 6) After Exp term3 power over factorial = 4/81 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore Exp cumulative sum through3 := sum(17/9, 4/81) After Exp cumulative sum through3 = 157/81 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore Exp term4 factorial step1 := mul(1, 1) After Exp term4 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Exp term4 factorial step2 := mul(1, 2) After Exp term4 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Exp term4 factorial step3 := mul(2, 3) After Exp term4 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Exp term4 factorial step4 := mul(6, 4) After Exp term4 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · impliesBefore (2/3)^4 After Exp ORIGINAL x power4 = 16/81 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S024 S026 · calculationBefore Exp term4 power over factorial := div(16/81, 24) After Exp term4 power over factorial = 2/243 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore Exp cumulative sum through4 := sum(157/81, 2/243) After Exp cumulative sum through4 = 473/243 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore Exp term5 factorial step1 := mul(1, 1) After Exp term5 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore Exp term5 factorial step2 := mul(1, 2) After Exp term5 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore Exp term5 factorial step3 := mul(2, 3) After Exp term5 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore Exp term5 factorial step4 := mul(6, 4) After Exp term5 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore Exp term5 factorial step5 := mul(24, 5) After Exp term5 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · impliesBefore (2/3)^5 After Exp ORIGINAL x power5 = 32/243 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S032 S034 · calculationBefore Exp term5 power over factorial := div(32/243, 120) After Exp term5 power over factorial = 4/3645 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore Exp cumulative sum through5 := sum(473/243, 4/3645) After Exp cumulative sum through5 = 7099/3645 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore Exp term6 factorial step1 := mul(1, 1) After Exp term6 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore Exp term6 factorial step2 := mul(1, 2) After Exp term6 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore Exp term6 factorial step3 := mul(2, 3) After Exp term6 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore Exp term6 factorial step4 := mul(6, 4) After Exp term6 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore Exp term6 factorial step5 := mul(24, 5) After Exp term6 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore Exp term6 factorial step6 := mul(120, 6) After Exp term6 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · impliesBefore (2/3)^6 After Exp ORIGINAL x power6 = 64/729 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S041 S043 · calculationBefore Exp term6 power over factorial := div(64/729, 720) After Exp term6 power over factorial = 4/32805 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore Exp cumulative sum through6 := sum(7099/3645, 4/32805) After Exp cumulative sum through6 = 12779/6561 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · impliesBefore source roles = [2/3, 6, [1, 2/3, 2/9, 4/81, 2/243, 4/3645, 4/32805], [1, 5/3, 17/9, 157/81, 473/243, 7099/3645, 12779/6561]] After derived = [12779/6561, No, finite sum is not a general infinite-value certificate] Conditions: none This calculation includes degree0 through the cutoff.No withholds a general exact infinite-value certificate.At x=0 the finite sum can equal E(0)=1,so do not assert universal inequality. Step identifiers and dependenciesRule: Exp_finite_sum_No_is_not_strict_inequality_at_every_x · Uses: G001, S044 ## Reverse checks - S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S044 · premise_and_result_audit — Recompute Exp cumulative sum through6 with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute Exp term6 power over factorial with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S041 · premise_and_result_audit — Recompute Exp term6 factorial step6 with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute Exp term6 factorial step5 with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute Exp term6 factorial step4 with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute Exp term6 factorial step3 with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute Exp term6 factorial step2 with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute Exp term6 factorial step1 with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute Exp cumulative sum through5 with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute Exp term5 power over factorial with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S032 · premise_and_result_audit — Recompute Exp term5 factorial step5 with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute Exp term5 factorial step4 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute Exp term5 factorial step3 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute Exp term5 factorial step2 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute Exp term5 factorial step1 with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute Exp cumulative sum through4 with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute Exp term4 power over factorial with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S024 · premise_and_result_audit — Recompute Exp term4 factorial step4 with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Exp term4 factorial step3 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Exp term4 factorial step2 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Exp term4 factorial step1 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute Exp cumulative sum through3 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Exp term3 power over factorial with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Recompute Exp term3 factorial step3 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Exp term3 factorial step2 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Exp term3 factorial step1 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Exp cumulative sum through2 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Exp term2 power over factorial with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Recompute Exp term2 factorial step2 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Exp term2 factorial step1 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Exp cumulative sum through1 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Exp term1 power over factorial with exact rational arithmetic, not rounded decimals. - 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 — Recompute Exp term1 factorial step1 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Exp cumulative sum through0 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Exp term0 power over factorial with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (2/3, 6, [1, 2/3, 2/9, 4/81, 2/243, 4/3645, 4/32805], [1, 5/3, 17/9, 157/81, 473/243, 7099/3645, 12779/6561], 12779/6561, No, finite_partial_sum_is_not_the_infinite_exponential_value) 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: 38123da9d4427ec0cef4a6a093af076547fa4511f7316958c9c438dafb08b24c - candidate record SHA-256: b9aa424902e8a7d799589b5c27ccf391c7be509a8e56e6790883c4d82fb3e714 - semantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386 - question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50b - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261113-r02-real-exponential-function-0002.json)