Question 23628: The One-Sided Tail Is a Limit of Finite-Cutoff Integrals

Original question and givens

Use f(x)=3/x^4 on [3,infinity), with integer exponent≥2 and lower≥1. Give (integrand, interval, integral, converges, certificate).

• G001 — Use f(x)=3/x^4 on [3,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

Step-by-step derivation

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

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

Improper ORIGINAL positive tail lower positive power step1 := mul(1, 3)

Improper ORIGINAL positive tail lower positive power step1 = 3

Improper ORIGINAL positive tail lower positive power step2 := mul(3, 3)

Improper ORIGINAL positive tail lower positive power step2 = 9

Improper ORIGINAL positive tail lower positive power step3 := mul(9, 3)

Improper ORIGINAL positive tail lower positive power step3 = 27

Improper ORIGINAL positive tail exponent-minus-one times lower-power := mul(3, 27)

Improper ORIGINAL positive tail exponent-minus-one times lower-power = 81

Improper ORIGINAL positive tail exact antiderivative limit value := div(3, 81)

Improper ORIGINAL positive tail exact antiderivative limit value = 1/27

Improper ORIGINAL positive tail:ORIGINAL scale=3,exponent=4>1,lower=3>=1;integrand scale/x^exponent

for EVERY B>=lower,integral_lower^B=scale/(exponent-1)*(lower^(1-exponent)-B^(1-exponent));B^(1-exponent) tends0,so limit=1/27

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 — Recompute Improper ORIGINAL positive tail exact antiderivative limit value with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Improper ORIGINAL positive tail exponent-minus-one times lower-power with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Improper ORIGINAL positive tail lower positive power step3 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Improper ORIGINAL positive tail lower positive power step2 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)=3/x^4, [3,infinity), 1/27, 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.