Home›Calculus 2›Radius of convergence›Problem 7.53

Radius of convergence

Problem 7.53 · easy

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

Use the ratio test on the absolute values of the terms.
\[ \lim_{n \to \infty}\left(\frac{x^{- n} x^{n + 1} n!}{x \left(n + 1\right)!}\right) = 0 \]
|a_(n+1)/a_n| → 0·|x − 0|.✓ Proved
The series converges when 0|x − 0| < 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 solution incorrectly states the convergence condition as 0|x - 0| < 1, which is nonsensical and implies a radius of 1 if interpreted literally, rather than correctly stating that the limit being 0 implies convergence for all x.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (misleading) 2026-09-28 — The solution incorrectly states the convergence condition as 0|x - 0| < 1, which is nonsensical and implies a radius of 1 if interpreted literally, rather than correctly stating that the limit being 0 implies convergence for all x.
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. The notation in step 3 is slightly informal but mathematically sound.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The ratio test limit is 0, which is strictly less than 1 for all x, implying convergence everywhere. The solution incorrectly states the condition as 0|x| < 1, which is logically false for |x| > 1 and misrepresents the test's conclusion.
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.