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Radius of convergence

Problem 7.109 · easy

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

Use the ratio test on the absolute values of the terms.
\[ \lim_{n \to \infty}\left(\frac{\left(x + 2\right)^{- n} \left(x + 2\right)^{n + 1} n!}{\left(x + 2\right) \left(n + 1\right)!}\right) = 0 \]
|a_(n+1)/a_n| → 0·|x − -2|.✓ Proved
The series converges when 0|x − -2| < 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 (style) — [domain objection, downgraded to style] The solution incorrectly applies the convergence condition for the Ratio Test. The limit of the ratio is 0, which is less than 1 for all x, so the series converges for all x. The solution erroneously sets up an inequality '0 < 1' as if the limit depended on x in a way that required solving for a radius, or implies the condition is |x+2| < 1/0 (undefined) rather than simply noting the limit is 0 < 1 for all x. The logic '0|x - -2| < 1' is mathematically nonsensical in this context because the |x+2| terms cancel out completely, leaving a limit of 0 independent of x.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-09-28 — [domain objection, downgraded to style] The solution incorrectly applies the convergence condition for the Ratio Test. The limit of the ratio is 0, which is less than 1 for all x, so the series converges for all x. The solution erroneously sets up an inequality '0 < 1' as if the limit depended on x in a way that required solving for a radius, or implies the condition is |x+2| < 1/0 (undefined) rather than simply noting the limit is 0 < 1 for all x. The logic '0|x - -2| < 1' is mathematically nonsensical in this context because the |x+2| terms cancel out completely, leaving a limit of 0 independent of x.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the Ratio Test. The limit of the ratio of consecutive terms is 0 for all x, implying the radius of convergence is infinity. The phrasing in step 3 is slightly informal but mathematically sound.
gpt-oss:20b: fail (misleading) 2026-09-27 — The conclusion sentence incorrectly states "0|x − -2| < 1"; it should simply note that the ratio limit is 0 for all x, hence the series converges for every real x. The wording misleads the reader into thinking a condition on x is required.
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution incorrectly applies the convergence condition for the ratio test. The condition is that the limit L must be less than 1. Since the limit is 0 for all x, the series converges for all x, but the reasoning '0|x + 2| < 1' implies a dependency on x that doesn't exist in the limit calculation (the limit is 0, not 0*|x+2|). More importantly, the standard ratio test limit for this series is 0, which is always < 1, so R is infinity. The phrasing '0|x - -2| < 1' is mathematically nonsensical as a condition derived from the limit being 0. The limit of the ratio is 0, not 0*|x+2|. The term |x+2| cancels out in the ratio a_{n+1}/a_n before taking the limit. Specifically, |a_{n+1}/a_n| = |x+2|/(n+1). The limit as n->inf is 0 for any fixed x. The condition is 0 < 1, which is always true. The solution's equation line shows the limit is 0, but the comment says it goes to 0*|x- -2|, which is confusing/wrong notation for the limit value. The sentence then uses this confused limit to derive a condition. It's better to say the limit is 0, which is < 1 for all x.
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.