Radius of convergence
Problem 7.144 · easy
Find the radius of convergence of \( \displaystyle \sum_{n=1}^{\infty} \frac{4^{- n} \left(x + 2\right)^{n}}{n^{2}} \).
- Use the ratio test on the absolute values of the terms.Reviewed
- \[ \lim_{n \to \infty}\left(\frac{4^{n} 4^{- n - 1} n^{2} \left(x + 2\right)^{- n} \left(x + 2\right)^{n + 1}}{\left(n + 1\right)^{2} \left(x + 2\right)}\right) = \frac{1}{4} \]|a_(n+1)/a_n| → 1/4·|x − -2|.✓ Proved
- The series converges when 1/4|x − -2| < 1, that is |x − -2| < 4.Reviewed
Answer \( R = 4 \)
✓ Nihil obstat Lines: 1 proved, 2 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 | 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 | 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 | terms evaluated just inside and just outside the radius shrink and grow |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the Ratio Test to determine the radius of convergence. The limit calculation and the resulting inequality are correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the Ratio Test to determine the radius of convergence. The limit calculation and the resulting inequality are correct.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the ratio test to determine the radius of convergence. The limit calculation and the resulting inequality are correct.gpt-oss:20b: pass 2026-09-26
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/radius_of_convergence, checked 2026-09-26 with SymPy 1.14.0.