Limits of sequences
Problem 7.458 · easy
Write the first four terms of \( \displaystyle a_n = \left(1 - \frac{3}{n}\right)^{n} \), and determine whether the sequence converges. If it does, find its limit.
- \[ \left[\begin{matrix}-2\\1 \cdot \frac{1}{4}\\0\\1 \cdot \frac{1}{256}\end{matrix}\right] = \left[\begin{matrix}-2\\\frac{1}{4}\\0\\\frac{1}{256}\end{matrix}\right] \]The first four terms.✓ Proved
- The limit (1 + c/n)ⁿ → eᶜ.Reviewed
- \[ \lim_{n \to \infty} \left(1 - \frac{3}{n}\right)^{n} = e^{-3} \]The limit.✓ Proved
Answer \( a_1, \dots, a_4 = -2, \frac{1}{4}, 0, \frac{1}{256};\ e^{-3} \)
✓ 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 applies the standard limit theorem for sequences of the form (1 + c/n)^n. The reasoning is sound and the conclusion is correct.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly identifies the first four terms and applies the standard limit theorem for sequences of the form (1 + c/n)^n. The reasoning is sound and the conclusion is correct.gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly identifies the first four terms and applies the standard limit formula lim (1 + c/n)^n = e^c with c = -3. The reasoning is sound and the results are correct.
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 2026-10-09 with SymPy 1.14.0.