The root test
Problem 7.730 · medium
Use the root test to decide whether \( \displaystyle \sum_{n=1}^{\infty} \left(\frac{4 n}{2 n + 4}\right)^{n} \) converges.
- The root test looks at lim |aₙ|^(1/n): below 1 converges, above 1 diverges.Reviewed
- \[ \left(\left(\frac{4 n}{2 n + 4}\right)^{n}\right)^{1 \frac{1}{n}} = \frac{4 n}{2 n + 4} \]The n-th root.✓ Proved
- \[ \lim_{n \to \infty}\left(\frac{4 n}{2 n + 4}\right) = 2 \]The limit is 2, > 1.✓ Proved
- So the series diverges.Reviewed
Answer \( \lim \sqrt[n]{|a_n|} = 2,\ \text{diverges} \)
Lines: 2 proved, 2 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. 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 2026-10-20 — The solution correctly applies the root test, computes the limit accurately, and draws the correct conclusion.gpt-oss:20b: pass 2026-10-20qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies the root test, computes the limit accurately, and draws the correct conclusion.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (error) 2026-10-19 — The limit calculation in step 3 is incorrect; the limit of 4n/(2n+4) as n approaches infinity is 2, not 2. Wait, 4n/2n is 2. The limit is indeed 2. Let me re-read carefully. 4n / (2n+4) -> 4/2 = 2. The limit is 2. 2 > 1, so it diverges. The solution is actually correct. Why did I think it was wrong? Ah, I misread the limit value in my head or something. Let's look closer. 4n/(2n+4) = 4/(2+4/n) -> 4/2 = 2. The limit is 2. The root test says if L > 1, it diverges. The solution says L=2, diverges. This is correct. Is there a subtle error? The problem asks to use the root test. The solution does. The algebra is checked by CAS. The logic is sound. Wait, is the limit 2? Yes. Is 2 > 1? Yes. Does it diverge? Yes. So why would this fail? Maybe the prompt implies I should find an error if there is one, but if there isn't, I pass. Let me double check the limit. lim n->inf (4n)/(2n+4) = 2. Correct. The root test condition: if lim sup |a_n|^(1/n) > 1, diverges. Here limit is 2. So it diverges. The solution is correct. I will pass it.gpt-oss:20b: pass 2026-10-19
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 2026-10-20 with SymPy 1.14.0.