Limits of sequences
Problem 7.377 · easy
Write the first four terms of \( \displaystyle a_n = \left(1 + \frac{1}{n}\right)^{n} \), and determine whether the sequence converges. If it does, find its limit.
- \[ \left[\begin{matrix}2\\\frac{9}{4}\\\frac{64}{27}\\\frac{625}{256}\end{matrix}\right] \]The first four terms.✓ Proved
- The limit (1 + c/n)ⁿ → eᶜ.Reviewed
- \[ \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n} = e \]The limit.✓ Proved
Answer \( a_1, \dots, a_4 = 2, \frac{9}{4}, \frac{64}{27}, \frac{625}{256};\ e \)
✓ 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 definition of e. The notation in sentence 2 is slightly informal but mathematically sound for this context.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the first four terms and applies the standard limit definition of e. The notation in sentence 2 is slightly informal but mathematically sound for this context.qwen3.6:27b-mlx: fail (style) 2026-10-06 — The solution is mathematically correct but stylistically poor. Step 1 presents a trivial matrix equality rather than explicitly listing the terms as requested. Step 2 cites a general theorem without explicitly connecting c=1 to the specific problem, and Step 3 relies on CAS output rather than a clear logical derivation from Step 2.gpt-oss:20b: pass 2026-10-06
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 2026-10-06 with SymPy 1.14.0.