Home›Calculus 2›Radius of convergence›Problem 7.163

Radius of convergence

Problem 7.163 · easy

Find the radius of convergence of \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(x + 2\right)^{n}}{n^{2}} \).

Use the ratio test on the absolute values of the terms.
Reviewed
\[ \lim_{n \to \infty}\left(\frac{n^{2} \left(x + 2\right)^{- n} \left(x + 2\right)^{n + 1}}{\left(n + 1\right)^{2} \left(x + 2\right)}\right) = 1 \]
|a_(n+1)/a_n| → 1·|x − -2|.✓ Proved
The series converges when 1|x − -2| < 1, that is |x − -2| < 1.
Reviewed

Answer \( R = 1 \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the ratio test to determine the radius of convergence. The algebraic simplification and the resulting inequality |x + 2| < 1 are correct, leading to the stated radius of 1.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the ratio test to determine the radius of convergence. The algebraic simplification and the resulting inequality |x + 2| < 1 are correct, leading to the stated radius of 1.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the ratio test to determine the radius of convergence. The limit calculation is correct, and the resulting inequality |x + 2| < 1 correctly identifies the radius as 1.
gpt-oss:20b: pass 2026-09-27

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 2026-09-27 with SymPy 1.14.0.