Limits of sequences
Problem 7.445 · easy
Write the first four terms of \( \displaystyle a_n = \left(1 - \frac{4}{n}\right)^{n} \), and determine whether the sequence converges. If it does, find its limit.
- \[ \left[\begin{matrix}-3\\1\\- \frac{1}{27}\\0\end{matrix}\right] \]The first four terms.✓ Proved
- The limit (1 + c/n)ⁿ → eᶜ.Reviewed
- \[ \lim_{n \to \infty} \left(1 - \frac{4}{n}\right)^{n} = e^{-4} \]The limit.✓ Proved
Answer \( a_1, \dots, a_4 = -3, 1, - \frac{1}{27}, 0;\ e^{-4} \)
Lines: 2 proved, 1 reviewed. 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 | 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 definition for e^c. The reasoning is sound and the results are correct.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the first four terms and applies the standard limit definition for e^c. The reasoning is sound and the results are correct.qwen3.6:27b-mlx: fail (error) 2026-10-08 — The first term a_1 is calculated as -3, but the formula yields (1 - 4/1)^1 = -3. Wait, let's re-read the problem. a_n = (1 - 4/n)^n. a_1 = (1-4)^1 = -3. a_2 = (1-2)^2 = 1. a_3 = (1-4/3)^3 = (-1/3)^3 = -1/27. a_4 = (1-1)^4 = 0. The terms are correct. The limit is e^-4. The solution is actually correct. Why did I think it was wrong? Let me double check the prompt's 'Stated answer'. Stated answer: a_1, ..., a_4 = -3, 1, -1/27, 0; e^-4. The solution matches the stated answer. The logic is sound. The sentence 'The limit (1 + c/n)^n -> e^c' is a standard theorem application. Is there any subtlety? For n=1, 2, 3, 4, the base is negative for n=1 and n=3. Does the limit theorem apply? Yes, the limit as n->infinity depends on the tail. The base approaches 1. The standard limit holds. The solution is correct.gpt-oss:20b: pass 2026-10-08
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 2026-10-08 with SymPy 1.14.0.