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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261007-advance-20261008-r01-sequence-limit-foundations · 2026-10-07 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
On natural indices starting at zero, u(n)=(-12)+(0)/(n+1), with d≥1. Find the declared reciprocal-tail limit.
-12
The reciprocal error tends to zero in this model, leaving L=-12, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
On natural indices starting at zero, u(n)=(-11)+(-2)/(n+2), with d≥1. Find the declared reciprocal-tail limit.
-11
The reciprocal error tends to zero in this model, leaving L=-11, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
On natural indices starting at zero, u(n)=(-10)+(3)/(n+3), with d≥1. Find the declared reciprocal-tail limit.
-10
The reciprocal error tends to zero in this model, leaving L=-10, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
On natural indices starting at zero, u(n)=(-9)+(-4)/(n+4), with d≥1. Find the declared reciprocal-tail limit.
-9
The reciprocal error tends to zero in this model, leaving L=-9, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
On natural indices starting at zero, u(n)=(-8)+(5)/(n+5), with d≥1. Find the declared reciprocal-tail limit.
-8
The reciprocal error tends to zero in this model, leaving L=-8, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
On natural indices starting at zero, u(n)=(-7)+(-6)/(n+1), with d≥1. Find the declared reciprocal-tail limit.
-7
The reciprocal error tends to zero in this model, leaving L=-7, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
On natural indices starting at zero, u(n)=(-6)+(7)/(n+2), with d≥1. Find the declared reciprocal-tail limit.
-6
The reciprocal error tends to zero in this model, leaving L=-6, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
On natural indices starting at zero, u(n)=(-5)+(-8)/(n+3), with d≥1. Find the declared reciprocal-tail limit.
-5
The reciprocal error tends to zero in this model, leaving L=-5, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
On natural indices starting at zero, u(n)=(-4)+(9)/(n+4), with d≥1. Find the declared reciprocal-tail limit.
-4
The reciprocal error tends to zero in this model, leaving L=-4, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
On natural indices starting at zero, u(n)=(-3)+(-10)/(n+5), with d≥1. Find the declared reciprocal-tail limit.
-3
The reciprocal error tends to zero in this model, leaving L=-3, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
On natural indices starting at zero, u(n)=(-2)+(11)/(n+1), with d≥1. Find the declared reciprocal-tail limit.
-2
The reciprocal error tends to zero in this model, leaving L=-2, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
On natural indices starting at zero, u(n)=(-1)+(-12)/(n+2), with d≥1. Find the declared reciprocal-tail limit.
-1
The reciprocal error tends to zero in this model, leaving L=-1, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
On natural indices starting at zero, u(n)=(0)+(13)/(n+3), with d≥1. Find the declared reciprocal-tail limit.
0
The reciprocal error tends to zero in this model, leaving L=0, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
On natural indices starting at zero, u(n)=(1)+(-14)/(n+4), with d≥1. Find the declared reciprocal-tail limit.
1
The reciprocal error tends to zero in this model, leaving L=1, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
On natural indices starting at zero, u(n)=(2)+(15)/(n+5), with d≥1. Find the declared reciprocal-tail limit.
2
The reciprocal error tends to zero in this model, leaving L=2, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
On natural indices starting at zero, u(n)=(3)+(-16)/(n+1), with d≥1. Find the declared reciprocal-tail limit.
3
The reciprocal error tends to zero in this model, leaving L=3, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
On natural indices starting at zero, u(n)=(4)+(17)/(n+2), with d≥1. Find the declared reciprocal-tail limit.
4
The reciprocal error tends to zero in this model, leaving L=4, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
On natural indices starting at zero, u(n)=(5)+(-18)/(n+3), with d≥1. Find the declared reciprocal-tail limit.
5
The reciprocal error tends to zero in this model, leaving L=5, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
On natural indices starting at zero, u(n)=(6)+(19)/(n+4), with d≥1. Find the declared reciprocal-tail limit.
6
The reciprocal error tends to zero in this model, leaving L=6, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
On natural indices starting at zero, u(n)=(7)+(-20)/(n+5), with d≥1. Find the declared reciprocal-tail limit.
7
The reciprocal error tends to zero in this model, leaving L=7, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
On natural indices starting at zero, u(n)=(8)+(21)/(n+1), with d≥1. Find the declared reciprocal-tail limit.
8
The reciprocal error tends to zero in this model, leaving L=8, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
On natural indices starting at zero, u(n)=(9)+(-22)/(n+2), with d≥1. Find the declared reciprocal-tail limit.
9
The reciprocal error tends to zero in this model, leaving L=9, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
On natural indices starting at zero, u(n)=(10)+(23)/(n+3), with d≥1. Find the declared reciprocal-tail limit.
10
The reciprocal error tends to zero in this model, leaving L=10, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
On natural indices starting at zero, u(n)=(11)+(-24)/(n+4), with d≥1. Find the declared reciprocal-tail limit.
11
The reciprocal error tends to zero in this model, leaving L=11, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
On natural indices starting at zero, u(n)=(12)+(25)/(n+5), with d≥1. Find the declared reciprocal-tail limit.
12
The reciprocal error tends to zero in this model, leaving L=12, including c=0. This is not an inference about an arbitrary sequence from finitely many terms.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-reciprocal-value · candidate
For an arbitrary fixed real L, u(n)=L+(0)/(n+1) on natural indices starting at zero, with d=1≥1. For ε=1/1>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
0
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=0. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(-2)/(n+2) on natural indices starting at zero, with d=2≥1. For ε=2/2>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
1
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=1. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(3)/(n+3) on natural indices starting at zero, with d=3≥1. For ε=3/3>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
1
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=1. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(-4)/(n+4) on natural indices starting at zero, with d=4≥1. For ε=4/1>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
0
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=0. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(5)/(n+1) on natural indices starting at zero, with d=1≥1. For ε=5/2>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
2
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=2. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(-6)/(n+2) on natural indices starting at zero, with d=2≥1. For ε=1/3>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
17
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=17. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(7)/(n+3) on natural indices starting at zero, with d=3≥1. For ε=2/1>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
1
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=1. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(-8)/(n+4) on natural indices starting at zero, with d=4≥1. For ε=3/2>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
2
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=2. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(9)/(n+1) on natural indices starting at zero, with d=1≥1. For ε=4/3>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
6
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=6. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(-10)/(n+2) on natural indices starting at zero, with d=2≥1. For ε=5/1>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
1
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=1. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(11)/(n+3) on natural indices starting at zero, with d=3≥1. For ε=1/2>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
20
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=20. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(-12)/(n+4) on natural indices starting at zero, with d=4≥1. For ε=2/3>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
15
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=15. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(13)/(n+1) on natural indices starting at zero, with d=1≥1. For ε=3/1>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
4
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=4. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(-14)/(n+2) on natural indices starting at zero, with d=2≥1. For ε=4/2>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
6
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=6. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(15)/(n+3) on natural indices starting at zero, with d=3≥1. For ε=5/3>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
7
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=7. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(-16)/(n+4) on natural indices starting at zero, with d=4≥1. For ε=1/1>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
13
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=13. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(17)/(n+1) on natural indices starting at zero, with d=1≥1. For ε=2/2>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
17
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=17. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(-18)/(n+2) on natural indices starting at zero, with d=2≥1. For ε=3/3>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
17
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=17. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(19)/(n+3) on natural indices starting at zero, with d=3≥1. For ε=4/1>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
2
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=2. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(-20)/(n+4) on natural indices starting at zero, with d=4≥1. For ε=5/2>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
5
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=5. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(21)/(n+1) on natural indices starting at zero, with d=1≥1. For ε=1/3>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
63
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=63. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(-22)/(n+2) on natural indices starting at zero, with d=2≥1. For ε=2/1>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
10
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=10. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(23)/(n+3) on natural indices starting at zero, with d=3≥1. For ε=3/2>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
13
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=13. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(-24)/(n+4) on natural indices starting at zero, with d=4≥1. For ε=4/3>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
15
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=15. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(25)/(n+1) on natural indices starting at zero, with d=1≥1. For ε=5/1>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.
5
The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=5. Equality fails the strict bound; a zero tail permits N=0.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-least-cutoff · candidate
For an arbitrary fixed real L, u(n)=L+(0)/(n+1) on natural indices starting at zero, with d=1≥1. With ε=1/1>0 and candidate N=0, does every n≥N have |u(n)−L|<ε, including n=N?
Yes
Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(-2)/(n+2) on natural indices starting at zero, with d=2≥1. With ε=2/2>0 and candidate N=0, does every n≥N have |u(n)−L|<ε, including n=N?
No
Check |c|q<p(N+d) exactly, giving No. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(3)/(n+3) on natural indices starting at zero, with d=3≥1. With ε=3/3>0 and candidate N=3, does every n≥N have |u(n)−L|<ε, including n=N?
Yes
Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(-4)/(n+4) on natural indices starting at zero, with d=4≥1. With ε=4/1>0 and candidate N=0, does every n≥N have |u(n)−L|<ε, including n=N?
Yes
Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(5)/(n+1) on natural indices starting at zero, with d=1≥1. With ε=5/2>0 and candidate N=1, does every n≥N have |u(n)−L|<ε, including n=N?
No
Check |c|q<p(N+d) exactly, giving No. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(-6)/(n+2) on natural indices starting at zero, with d=2≥1. With ε=1/3>0 and candidate N=19, does every n≥N have |u(n)−L|<ε, including n=N?
Yes
Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(7)/(n+3) on natural indices starting at zero, with d=3≥1. With ε=2/1>0 and candidate N=1, does every n≥N have |u(n)−L|<ε, including n=N?
Yes
Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(-8)/(n+4) on natural indices starting at zero, with d=4≥1. With ε=3/2>0 and candidate N=1, does every n≥N have |u(n)−L|<ε, including n=N?
No
Check |c|q<p(N+d) exactly, giving No. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(9)/(n+1) on natural indices starting at zero, with d=1≥1. With ε=4/3>0 and candidate N=8, does every n≥N have |u(n)−L|<ε, including n=N?
Yes
Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(-10)/(n+2) on natural indices starting at zero, with d=2≥1. With ε=5/1>0 and candidate N=1, does every n≥N have |u(n)−L|<ε, including n=N?
Yes
Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(11)/(n+3) on natural indices starting at zero, with d=3≥1. With ε=1/2>0 and candidate N=19, does every n≥N have |u(n)−L|<ε, including n=N?
No
Check |c|q<p(N+d) exactly, giving No. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(-12)/(n+4) on natural indices starting at zero, with d=4≥1. With ε=2/3>0 and candidate N=17, does every n≥N have |u(n)−L|<ε, including n=N?
Yes
Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(13)/(n+1) on natural indices starting at zero, with d=1≥1. With ε=3/1>0 and candidate N=4, does every n≥N have |u(n)−L|<ε, including n=N?
Yes
Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(-14)/(n+2) on natural indices starting at zero, with d=2≥1. With ε=4/2>0 and candidate N=5, does every n≥N have |u(n)−L|<ε, including n=N?
No
Check |c|q<p(N+d) exactly, giving No. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(15)/(n+3) on natural indices starting at zero, with d=3≥1. With ε=5/3>0 and candidate N=9, does every n≥N have |u(n)−L|<ε, including n=N?
Yes
Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(-16)/(n+4) on natural indices starting at zero, with d=4≥1. With ε=1/1>0 and candidate N=13, does every n≥N have |u(n)−L|<ε, including n=N?
Yes
Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(17)/(n+1) on natural indices starting at zero, with d=1≥1. With ε=2/2>0 and candidate N=16, does every n≥N have |u(n)−L|<ε, including n=N?
No
Check |c|q<p(N+d) exactly, giving No. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(-18)/(n+2) on natural indices starting at zero, with d=2≥1. With ε=3/3>0 and candidate N=19, does every n≥N have |u(n)−L|<ε, including n=N?
Yes
Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(19)/(n+3) on natural indices starting at zero, with d=3≥1. With ε=4/1>0 and candidate N=2, does every n≥N have |u(n)−L|<ε, including n=N?
Yes
Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(-20)/(n+4) on natural indices starting at zero, with d=4≥1. With ε=5/2>0 and candidate N=4, does every n≥N have |u(n)−L|<ε, including n=N?
No
Check |c|q<p(N+d) exactly, giving No. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(21)/(n+1) on natural indices starting at zero, with d=1≥1. With ε=1/3>0 and candidate N=65, does every n≥N have |u(n)−L|<ε, including n=N?
Yes
Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(-22)/(n+2) on natural indices starting at zero, with d=2≥1. With ε=2/1>0 and candidate N=10, does every n≥N have |u(n)−L|<ε, including n=N?
Yes
Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(23)/(n+3) on natural indices starting at zero, with d=3≥1. With ε=3/2>0 and candidate N=12, does every n≥N have |u(n)−L|<ε, including n=N?
No
Check |c|q<p(N+d) exactly, giving No. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(-24)/(n+4) on natural indices starting at zero, with d=4≥1. With ε=4/3>0 and candidate N=17, does every n≥N have |u(n)−L|<ε, including n=N?
Yes
Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(25)/(n+1) on natural indices starting at zero, with d=1≥1. With ε=5/1>0 and candidate N=5, does every n≥N have |u(n)−L|<ε, including n=N?
Yes
Check |c|q<p(N+d) exactly, giving Yes. Equality is not strict enough. The all-tail reduction uses this model's nonincreasing error, not sampling as a proof for an arbitrary sequence.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-cutoff-validity · candidate
For an arbitrary fixed real L, u(n)=L+(0)/(n+1) on natural indices starting at zero, with d=1≥1. Find the exact absolute error |u(0)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
0/1
Use |c|/(n+d) to obtain 0/1. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
For an arbitrary fixed real L, u(n)=L+(-2)/(n+2) on natural indices starting at zero, with d=2≥1. Find the exact absolute error |u(1)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
2/3
Use |c|/(n+d) to obtain 2/3. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
For an arbitrary fixed real L, u(n)=L+(3)/(n+3) on natural indices starting at zero, with d=3≥1. Find the exact absolute error |u(2)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
3/5
Use |c|/(n+d) to obtain 3/5. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
For an arbitrary fixed real L, u(n)=L+(-4)/(n+4) on natural indices starting at zero, with d=4≥1. Find the exact absolute error |u(3)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
4/7
Use |c|/(n+d) to obtain 4/7. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
For an arbitrary fixed real L, u(n)=L+(5)/(n+1) on natural indices starting at zero, with d=1≥1. Find the exact absolute error |u(4)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
1/1
Use |c|/(n+d) to obtain 1/1. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
For an arbitrary fixed real L, u(n)=L+(-6)/(n+2) on natural indices starting at zero, with d=2≥1. Find the exact absolute error |u(5)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
6/7
Use |c|/(n+d) to obtain 6/7. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
For an arbitrary fixed real L, u(n)=L+(7)/(n+3) on natural indices starting at zero, with d=3≥1. Find the exact absolute error |u(6)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
7/9
Use |c|/(n+d) to obtain 7/9. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
For an arbitrary fixed real L, u(n)=L+(-8)/(n+4) on natural indices starting at zero, with d=4≥1. Find the exact absolute error |u(0)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
2/1
Use |c|/(n+d) to obtain 2/1. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
For an arbitrary fixed real L, u(n)=L+(9)/(n+1) on natural indices starting at zero, with d=1≥1. Find the exact absolute error |u(1)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
9/2
Use |c|/(n+d) to obtain 9/2. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
For an arbitrary fixed real L, u(n)=L+(-10)/(n+2) on natural indices starting at zero, with d=2≥1. Find the exact absolute error |u(2)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
5/2
Use |c|/(n+d) to obtain 5/2. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
For an arbitrary fixed real L, u(n)=L+(11)/(n+3) on natural indices starting at zero, with d=3≥1. Find the exact absolute error |u(3)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
11/6
Use |c|/(n+d) to obtain 11/6. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
For an arbitrary fixed real L, u(n)=L+(-12)/(n+4) on natural indices starting at zero, with d=4≥1. Find the exact absolute error |u(4)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
3/2
Use |c|/(n+d) to obtain 3/2. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
For an arbitrary fixed real L, u(n)=L+(13)/(n+1) on natural indices starting at zero, with d=1≥1. Find the exact absolute error |u(5)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
13/6
Use |c|/(n+d) to obtain 13/6. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
For an arbitrary fixed real L, u(n)=L+(-14)/(n+2) on natural indices starting at zero, with d=2≥1. Find the exact absolute error |u(6)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
7/4
Use |c|/(n+d) to obtain 7/4. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
For an arbitrary fixed real L, u(n)=L+(15)/(n+3) on natural indices starting at zero, with d=3≥1. Find the exact absolute error |u(0)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
5/1
Use |c|/(n+d) to obtain 5/1. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
For an arbitrary fixed real L, u(n)=L+(-16)/(n+4) on natural indices starting at zero, with d=4≥1. Find the exact absolute error |u(1)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
16/5
Use |c|/(n+d) to obtain 16/5. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
For an arbitrary fixed real L, u(n)=L+(17)/(n+1) on natural indices starting at zero, with d=1≥1. Find the exact absolute error |u(2)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
17/3
Use |c|/(n+d) to obtain 17/3. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
For an arbitrary fixed real L, u(n)=L+(-18)/(n+2) on natural indices starting at zero, with d=2≥1. Find the exact absolute error |u(3)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
18/5
Use |c|/(n+d) to obtain 18/5. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
For an arbitrary fixed real L, u(n)=L+(19)/(n+3) on natural indices starting at zero, with d=3≥1. Find the exact absolute error |u(4)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
19/7
Use |c|/(n+d) to obtain 19/7. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
For an arbitrary fixed real L, u(n)=L+(-20)/(n+4) on natural indices starting at zero, with d=4≥1. Find the exact absolute error |u(5)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
20/9
Use |c|/(n+d) to obtain 20/9. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
For an arbitrary fixed real L, u(n)=L+(21)/(n+1) on natural indices starting at zero, with d=1≥1. Find the exact absolute error |u(6)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
3/1
Use |c|/(n+d) to obtain 3/1. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
For an arbitrary fixed real L, u(n)=L+(-22)/(n+2) on natural indices starting at zero, with d=2≥1. Find the exact absolute error |u(0)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
11/1
Use |c|/(n+d) to obtain 11/1. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
For an arbitrary fixed real L, u(n)=L+(23)/(n+3) on natural indices starting at zero, with d=3≥1. Find the exact absolute error |u(1)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
23/4
Use |c|/(n+d) to obtain 23/4. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
For an arbitrary fixed real L, u(n)=L+(-24)/(n+4) on natural indices starting at zero, with d=4≥1. Find the exact absolute error |u(2)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
4/1
Use |c|/(n+d) to obtain 4/1. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
For an arbitrary fixed real L, u(n)=L+(25)/(n+1) on natural indices starting at zero, with d=1≥1. Find the exact absolute error |u(3)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.
25/4
Use |c|/(n+d) to obtain 25/4. Retain nonnegative zero-based indexing and the positive denominator; changing L does not affect this error.
Reviewed step-by-step derivation
Sequence limits and exact tail cutoffs · qf-sequence-limit-term-error · candidate
Canonical question JSONL · English JSONL · Validation report