Limits of sequences
Problem 7.297 · easy
Write the first four terms of \( \displaystyle a_n = n \sin{\left(\frac{1}{n} \right)} \), and determine whether the sequence converges. If it does, find its limit.
- \[ \left[\begin{matrix}\sin{\left(1 \right)}\\2 \sin{\left(1 \cdot \frac{1}{2} \right)}\\3 \sin{\left(1 \cdot \frac{1}{3} \right)}\\4 \sin{\left(1 \cdot \frac{1}{4} \right)}\end{matrix}\right] = \left[\begin{matrix}\sin{\left(1 \right)}\\2 \sin{\left(\frac{1}{2} \right)}\\3 \sin{\left(\frac{1}{3} \right)}\\4 \sin{\left(\frac{1}{4} \right)}\end{matrix}\right] \]The first four terms.✓ Proved
- With x = 1/n this is sin(x)/x as x → 0⁺.Reviewed
- \[ \lim_{n \to \infty}\left(n \sin{\left(\frac{1}{n} \right)}\right) = 1 \]The limit.✓ Proved
Answer \( a_1, \dots, a_4 = \sin{\left(1 \right)}, 2 \sin{\left(\frac{1}{2} \right)}, 3 \sin{\left(\frac{1}{3} \right)}, 4 \sin{\left(\frac{1}{4} \right)};\ 1 \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the first four terms and accurately applies the standard limit lim_{x->0} sin(x)/x = 1 via the substitution x = 1/n to determine convergence.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the first four terms and accurately applies the standard limit lim_{x->0} sin(x)/x = 1 via the substitution x = 1/n to determine convergence.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the first four terms and accurately determines the limit using the standard substitution x = 1/n, which transforms the expression into the well-known limit sin(x)/x as x approaches 0.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.