Home›Calculus 2›Radius of convergence›Problem 7.55

Radius of convergence

Problem 7.55 · 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. 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	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 limit calculation is correct, and the resulting inequality correctly identifies the radius as 1.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the ratio test to determine the radius of convergence. The limit calculation is correct, and the resulting inequality correctly identifies the radius as 1.
gpt-oss:20b: pass 2026-09-28
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
qwen3.6:27b-mlx: pass 2026-09-26 — 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 correct radius R = 1.
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 2026-09-28 with SymPy 1.14.0.