Limits of sequences
Problem 7.298 · easy
Write the first four terms of \( \displaystyle a_n = \frac{\ln{\left(n \right)}}{n} \), and determine whether the sequence converges. If it does, find its limit.
- \[ \left[\begin{matrix}0\\\frac{\ln{\left(2 \right)}}{2}\\\frac{\ln{\left(3 \right)}}{3}\\\frac{\ln{\left(4 \right)}}{4}\end{matrix}\right] = \left[\begin{matrix}0\\\frac{\ln{\left(2 \right)}}{2}\\\frac{\ln{\left(3 \right)}}{3}\\\frac{\ln{\left(2 \right)}}{2}\end{matrix}\right] \]The first four terms.✓ Proved
- L'Hôpital on ln(x)/x.
- \[ \lim_{n \to \infty}\left(\frac{\ln{\left(n \right)}}{n}\right) = 0 \]The limit.✓ Proved
Answer \( a_1, \dots, a_4 = 0, \frac{\ln{\left(2 \right)}}{2}, \frac{\ln{\left(3 \right)}}{3}, \frac{\ln{\left(2 \right)}}{2};\ 0 \)
Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the terms evaluated at n = 10⁶, 10⁶ + 1 and 10⁷ |
Reviewers
gpt-oss:20b: fail (error) — The fourth term is incorrectly given as ln(2)/2 instead of ln(4)/4, and the sentence claiming “L’Hôpital on ln(x)/x” is vague and does not justify the limit or convergence of the sequence.qwen3.6:27b-mlx: fail (error) — The stated answer lists the fourth term as ln(2)/2, which is incorrect; it should be ln(4)/4. The solution's equation line attempts to assert this incorrect equality, and while marked 'proved' by a CAS, the CAS output clearly shows log(4)/4 on the LHS, making the equality false. The solution fails to provide the correct terms.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The stated answer lists the fourth term as ln(2)/2, which is incorrect; it should be ln(4)/4. The solution's equation line attempts to assert this incorrect equality, and while marked 'proved' by a CAS, the CAS output clearly shows log(4)/4 on the LHS, making the equality false. The solution fails to provide the correct terms.gpt-oss:20b: fail (error) 2026-10-05 — The fourth term is incorrectly given as ln(2)/2 instead of ln(4)/4, and the sentence claiming “L’Hôpital on ln(x)/x” is vague and does not justify the limit or convergence of the sequence.qwen3.6:27b-mlx: fail (error) 2026-10-05 — The stated answer contains a typo in the fourth term, listing ln(2)/2 instead of ln(4)/4. The solution's first equation is mathematically false (log(3)/3 does not equal log(2)/2), and the CAS check failed to flag this inconsistency.gpt-oss:20b: pass 2026-10-05
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/sequence_limits, checked 2026-10-05 with SymPy 1.14.0.