# Question 23646: The One-Sided Tail Is a Limit of Finite-Cutoff Integrals

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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.

R0046-23646 · q-20261014-advance-20261112-r04-improper-integral-0021



## Original question and givens

Use f(x)=21/x^2 on [1,infinity), with integer exponent≥2 and lower≥1. Give (integrand, interval, integral, converges, certificate).


- G001 — Use f(x)=21/x^2 on [1,infinity), with integer exponent≥2 and lower≥1. Give (integrand, interval, integral, converges, 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-improper-integral](https://ocme.evemisslab.com/en/mko/mko-improper-integral/)





## Step-by-step derivation

S001 · impliesBefore

source operation=improper-one-sided-tail;original snapshot and mathematics retained

After

explicit ordered source output fields=[integrand, interval, integral, converges, certificate]

Conditions: none

Keep the source snapshot while listing ordered fields without JavaScript.Radius uses its original mode to select6 or11 fields without padding;distinguish nonclaims from actual negations.

Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · calculationBefore

Improper ORIGINAL positive tail lower positive power step1 := mul(1, 1)

After

Improper ORIGINAL positive tail lower positive power step1 = 1

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

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





S003 · calculationBefore

Improper ORIGINAL positive tail exponent-minus-one times lower-power := mul(1, 1)

After

Improper ORIGINAL positive tail exponent-minus-one times lower-power = 1

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

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





S004 · calculationBefore

Improper ORIGINAL positive tail exact antiderivative limit value := div(21, 1)

After

Improper ORIGINAL positive tail exact antiderivative limit value = 21

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

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





S005 · impliesBefore

Improper ORIGINAL positive tail:ORIGINAL scale=21,exponent=2>1,lower=1>=1;integrand scale/x^exponent

After

for EVERY B>=lower,integral_lower^B=scale/(exponent-1)*(lower^(1-exponent)-B^(1-exponent));B^(1-exponent) tends0,so limit=21

Conditions: none

Because k>1,the negative power tends0.The absolute tail error is |scale|/((k−1)B^(k−1)),which is below any epsilon for sufficiently large B.Signed or zero scale is allowed;this is a universal limit,not sampled cutoffs.

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









## Reverse checks


- 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 — Recompute Improper ORIGINAL positive tail exact antiderivative limit value with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute Improper ORIGINAL positive tail exponent-minus-one times lower-power with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute Improper ORIGINAL positive tail lower positive power step1 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

(f(x)=21/x^2, [1,infinity), 21, Yes, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)

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: 2e0727089b136fe0f84209cb8628bf675de1c962bfc0fadc0e611ba70fe2d4a2
- candidate record SHA-256: 05f03fa571897dedf4c08642b2c9787264c0f75e36d813ed7ac2511aec0f5227
- semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1
- question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261112-r04-improper-integral-0021.json)