The root test
Problem 7.641 · medium
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{n}{n + 1}\right)^{n^{2}} \) converges.
- The root test looks at lim |aₙ|^(1/n): below 1 converges, above 1 diverges.Reviewed
- \[ \left(\left(\frac{n}{n + 1}\right)^{n^{2}}\right)^{1 \frac{1}{n}} = \left(\frac{n}{n + 1}\right)^{n} \]The n-th root.✓ Proved
- \[ \lim_{n \to \infty} \left(\frac{n}{n + 1}\right)^{n} = e^{-1} \]The limit is exp(-1), < 1.✓ Proved
- So the series converges.Reviewed
Answer \( \lim \sqrt[n]{|a_n|} = e^{-1},\ \text{converges} \)
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | |aₙ|^(1/n) evaluated at n = 10⁶ in 60-digit arithmetic |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the root test, computes the limit accurately, and draws the correct conclusion.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies the root test, computes the limit accurately, and draws the correct conclusion.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the root test, computes the limit accurately, and draws the correct conclusion.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the root test, computes the limit accurately, and draws the correct conclusion.gpt-oss:20b: pass 2026-10-15
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/root_test, checked 2026-10-18 with SymPy 1.14.0.