Question 23676: Two Infinite Tails Must Converge Separately before Their Values Are Added

Original question and givens

Use f(x)=1/abs(x)^2 for x≤−1, f(x)=0 for −1<x<1 and f(x)=2/x^2 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)=1/abs(x)^2 for x≤−1, f(x)=0 for −1<x<1 and f(x)=2/x^2 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 exponent-minus-one times lower-power := mul(1, 1)

Improper ORIGINAL negative tail after x=-u exponent-minus-one times lower-power = 1

Improper ORIGINAL negative tail after x=-u exact antiderivative limit value := div(1, 1)

Improper ORIGINAL negative tail after x=-u exact antiderivative limit value = 1

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

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

Improper ORIGINAL positive tail lower positive power step1 = 1

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

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

Improper ORIGINAL positive tail exact antiderivative limit value := div(2, 1)

Improper ORIGINAL positive tail exact antiderivative limit value = 2

Improper ORIGINAL positive tail:ORIGINAL scale=2,exponent=2>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 independently convergent tails plus zero middle := sum(1, 2)

Improper independently convergent tails plus zero middle = 3

ORIGINAL piecewise scales left=1,right=2,exponent=2;middle(-1,1) integrand0

negative-tail limit=1;positive-tail limit=2;BOTH independently finite,then ordinary sum=3

Reverse checks

• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Recompute Improper independently convergent tails plus zero middle with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Recompute Improper ORIGINAL positive tail exact antiderivative limit value with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Improper ORIGINAL positive tail exponent-minus-one times lower-power with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Improper ORIGINAL positive tail lower positive power step1 with exact rational arithmetic, not rounded decimals.
• 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 negative tail after x=-u exact antiderivative limit value with exact rational arithmetic, not rounded decimals.
• S003 · 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.
• 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<=-1:1/abs(x)^2;-1<x<1:0;x>=1:2/x^2, (-infinity,infinity), 1, 2, Yes, Yes, 3, 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.