Home›Calculus 2›Radius of convergence›Problem 7.111

Radius of convergence

Problem 7.111 · easy

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

Use the ratio test on the absolute values of the terms.
Reviewed
\[ \lim_{n \to \infty}\left(\frac{\left(x - 1\right)^{- n} \left(x - 1\right)^{n + 1} n!}{\left(x - 1\right) \left(n + 1\right)!}\right) = 0 \]
|a_(n+1)/a_n| → 0·|x − 1|.✓ Proved
The series converges when 0|x − 1| < 1, which is every x: R = ∞.
Reviewed

Answer \( R = \infty \)

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 and derives the correct radius of convergence.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the ratio test and derives the correct radius of convergence.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the ratio test and derives the correct radius of convergence.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution incorrectly applies the ratio test condition for convergence (L < 1) to a limit that is 0. Since the limit is 0 for all finite x, the series converges for all x, but the reasoning '0|x-1| < 1' is logically flawed because the limit is 0, not 0|x-1|, and the condition L < 1 is satisfied trivially for all x, not just because 0 < 1. More importantly, the limit calculation in step 2 is presented as 0, but the text says '|a_(n+1)/a_n| -> 0*|x-1|', which is confusing notation. The limit of the ratio is actually 0 for any fixed x. The conclusion R=infinity is correct, but the justification is garbled and technically incorrect regarding the limit value.
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.