Question 23683: Two Infinite Tails Must Converge Separately before Their Values Are Added

Original question and givens

Use f(x)=8/abs(x)^5 for x≤−1, f(x)=0 for −1<x<1 and f(x)=9/x^5 for x≥1, with integer exponent≥2. Give (piecewise_integrand, interval, left_tail_integral, right_tail_integral, left_converges, right_converges, sum, certificate).

• G001 — Use f(x)=8/abs(x)^5 for x≤−1, f(x)=0 for −1<x<1 and f(x)=9/x^5 for x≥1, with integer exponent≥2. Give (piecewise_integrand, interval, left_tail_integral, right_tail_integral, left_converges, right_converges, sum, 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-two-sided-independent;original snapshot and mathematics retained

explicit ordered source output fields=[piecewise_integrand, interval, left_tail_integral, right_tail_integral, left_converges, right_converges, sum, certificate]

Improper ORIGINAL negative tail after x=-u lower positive power step1 := mul(1, 1)

Improper ORIGINAL negative tail after x=-u lower positive power step1 = 1

Improper ORIGINAL negative tail after x=-u lower positive power step2 := mul(1, 1)

Improper ORIGINAL negative tail after x=-u lower positive power step2 = 1

Improper ORIGINAL negative tail after x=-u lower positive power step3 := mul(1, 1)

Improper ORIGINAL negative tail after x=-u lower positive power step3 = 1

Improper ORIGINAL negative tail after x=-u lower positive power step4 := mul(1, 1)

Improper ORIGINAL negative tail after x=-u lower positive power step4 = 1

Improper ORIGINAL negative tail after x=-u exponent-minus-one times lower-power := mul(4, 1)

Improper ORIGINAL negative tail after x=-u exponent-minus-one times lower-power = 4

Improper ORIGINAL negative tail after x=-u exact antiderivative limit value := div(8, 4)

Improper ORIGINAL negative tail after x=-u exact antiderivative limit value = 2

Improper ORIGINAL negative tail after x=-u:ORIGINAL scale=8,exponent=5>1,lower=1>=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=2

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

Improper ORIGINAL positive tail lower positive power step1 = 1

Improper ORIGINAL positive tail lower positive power step2 := mul(1, 1)

Improper ORIGINAL positive tail lower positive power step2 = 1

Improper ORIGINAL positive tail lower positive power step3 := mul(1, 1)

Improper ORIGINAL positive tail lower positive power step3 = 1

Improper ORIGINAL positive tail lower positive power step4 := mul(1, 1)

Improper ORIGINAL positive tail lower positive power step4 = 1

Improper ORIGINAL positive tail exponent-minus-one times lower-power := mul(4, 1)

Improper ORIGINAL positive tail exponent-minus-one times lower-power = 4

Improper ORIGINAL positive tail exact antiderivative limit value := div(9, 4)

Improper ORIGINAL positive tail exact antiderivative limit value = 9/4

Improper ORIGINAL positive tail:ORIGINAL scale=9,exponent=5>1,lower=1>=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=9/4

Improper independently convergent tails plus zero middle := sum(2, 9/4)

Improper independently convergent tails plus zero middle = 17/4

ORIGINAL piecewise scales left=8,right=9,exponent=5;middle(-1,1) integrand0

negative-tail limit=2;positive-tail limit=9/4;BOTH independently finite,then ordinary sum=17/4

Reverse checks

• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Recompute Improper independently convergent tails plus zero middle with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Recompute Improper ORIGINAL positive tail exact antiderivative limit value with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Improper ORIGINAL positive tail exponent-minus-one times lower-power with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Improper ORIGINAL positive tail lower positive power step4 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Improper ORIGINAL positive tail lower positive power step3 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Improper ORIGINAL positive tail lower positive power step2 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Improper ORIGINAL positive tail lower positive power step1 with exact rational arithmetic, not rounded decimals.
• 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 — Recompute Improper ORIGINAL negative tail after x=-u exact antiderivative limit value with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Improper ORIGINAL negative tail after x=-u exponent-minus-one times lower-power with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Improper ORIGINAL negative tail after x=-u lower positive power step4 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Improper ORIGINAL negative tail after x=-u lower positive power step3 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Improper ORIGINAL negative tail after x=-u lower positive power step2 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Improper ORIGINAL negative tail after x=-u 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

(x<=-1:8/abs(x)^5;-1<x<1:0;x>=1:9/x^5, (-infinity,infinity), 2, 9/4, Yes, Yes, 17/4, both_one_sided_tail_integrals_converge_independently)

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.