Home›Calculus 2›Radius of convergence›Problem 7.51

Radius of convergence

Problem 7.51 · easy

Find the radius of convergence of \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(x - 3\right)^{n}}{n!} \).

Use the ratio test on the absolute values of the terms.
\[ \lim_{n \to \infty}\left(\frac{\left(x - 3\right)^{- n} \left(x - 3\right)^{n + 1} n!}{\left(x - 3\right) \left(n + 1\right)!}\right) = 0 \]
|a_(n+1)/a_n| → 0·|x − 3|.✓ Proved
The series converges when 0|x − 3| < 1, which is every x: R = ∞.

Answer \( R = \infty \)

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 (misleading) — The explanation in line 3 is logically incoherent. It claims convergence when '0|x-3| < 1', but 0|x-3| is always 0, so the condition is trivially true for all x. While the final answer R=∞ is correct, the reasoning presented is nonsense and would teach a student that the ratio test limit being 0 implies a condition '0 < 1' rather than understanding that L=0 < 1 for all x.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (misleading) 2026-09-28 — The explanation in line 3 is logically incoherent. It claims convergence when '0|x-3| < 1', but 0|x-3| is always 0, so the condition is trivially true for all x. While the final answer R=∞ is correct, the reasoning presented is nonsense and would teach a student that the ratio test limit being 0 implies a condition '0 < 1' rather than understanding that L=0 < 1 for all x.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution incorrectly applies the convergence condition for the Ratio Test. It states the series converges when the limit is less than 1, but the limit is 0, which is always less than 1 regardless of x. The condition '0|x - 3| < 1' is mathematically nonsensical and implies a bounded radius, whereas the correct conclusion is that the limit being 0 implies convergence for all x.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the ratio test and derives the correct radius of convergence.
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.