Home›Calculus 2›Radius of convergence›Problem 7.211

Radius of convergence

Problem 7.211 · easy

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

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

Answer \( R = 1 \)

Lines: 1 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	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 final inequality are correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the ratio test to determine the radius of convergence. The algebraic simplification and final inequality are correct.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies the ratio test to determine the radius of convergence. The algebraic simplification and final inequality are correct.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: fail (error) 2026-09-29 — The solution incorrectly applies the Ratio Test by simplifying the ratio of terms to 1, ignoring the factor of (x+2). The limit of |a_{n+1}/a_n| is |x+2|, not 1. Consequently, the condition for convergence is derived incorrectly.
gpt-oss:20b: pass 2026-09-29

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 2026-10-04 with SymPy 1.14.0.