Home›Calculus 2›Radius of convergence›Problem 7.52

Radius of convergence

Problem 7.52 · 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.
\[ \lim_{n \to \infty}\left(\frac{\left(x + 1\right)^{- n} \left(x + 1\right)^{n + 1} n!}{\left(x + 1\right) \left(n + 1\right)!}\right) = 0 \]
|a_(n+1)/a_n| → 0·|x − -1|.✓ Proved
The series converges when 0|x − -1| < 1, which is every x: R = ∞.

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 (error) — 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, implying convergence for all x. While the final answer is correct, the reasoning '0|x - -1| < 1' is mathematically nonsensical and misrepresents the test (which requires the limit of the ratio of coefficients to be 0, not multiplied by |x-c| in the inequality check in this manner). A clearer explanation would note that since the limit of |a_{n+1}/a_n| is 0 for any fixed x, the series converges everywhere.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — 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, implying convergence for all x. While the final answer is correct, the reasoning '0|x - -1| < 1' is mathematically nonsensical and misrepresents the test (which requires the limit of the ratio of coefficients to be 0, not multiplied by |x-c| in the inequality check in this manner). A clearer explanation would note that since the limit of |a_{n+1}/a_n| is 0 for any fixed x, the series converges everywhere.
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: pass 2026-09-26 — The solution correctly applies the ratio test. Although the intermediate notation in step 2 is slightly garbled, the limit calculation is correct (limit is 0), and the conclusion that the radius of convergence is infinity follows logically.
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.