Home›Calculus 2›Radius of convergence›Problem 7.168

Radius of convergence

Problem 7.168 · easy

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

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

Answer \( R = 2 \)

Lines: 1 proved, 2 not checked. 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	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	Not checked	—	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: fail (error) — The limit calculation in step 2 is incorrect; the ratio of consecutive terms is 2^(-1)(x+1), so the limit is |x+1|/2, not 1/2. The text annotation claiming the limit is 1/2 * |x - -1| is also mathematically inconsistent with the computed limit of 1/2.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The limit calculation in step 2 is incorrect; the ratio of consecutive terms is 2^(-1)(x+1), so the limit is |x+1|/2, not 1/2. The text annotation claiming the limit is 1/2 * |x - -1| is also mathematically inconsistent with the computed limit of 1/2.
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 algebraic simplification and final inequality are correct.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the ratio test to find the radius of convergence. The algebraic simplification and final inequality are correct.
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 2026-09-28 with SymPy 1.14.0.