Home›Calculus 2›Radius of convergence›Problem 7.214

Radius of convergence

Problem 7.214 · easy

Find the radius of convergence of $ \displaystyle \sum_{n=1}^{\infty} x^{n} $.

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

Answer $ R = 1 $

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.

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 Ratio Test by including the variable $x$ in the limit as $n \to \infty$. The Ratio Test requires taking the limit of the ratio of coefficients $|a_{n+1}/a_n|$, which is 1, independent of $x$. The condition for convergence is then $L|x| < 1$, not that the limit itself depends on $x$.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-09-29 — The solution incorrectly applies the Ratio Test by including the variable $x$ in the limit as $n \to \infty$. The Ratio Test requires taking the limit of the ratio of coefficients $|a_{n+1}/a_n|$, which is 1, independent of $x$. The condition for convergence is then $L|x| < 1$, not that the limit itself depends on $x$.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: fail (error) 2026-09-29 — The solution incorrectly applies the ratio test to the expression x^(n+1)/(x*x^n), which simplifies to 1/x, rather than the ratio of consecutive terms a_{n+1}/a_n = x. This leads to the nonsensical condition |x - 0| < 1 derived from a limit of 1/|x|, confusing the variable x with the center of the series or the limit value.
gpt-oss:20b: pass 2026-09-29

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 2026-09-29 with SymPy 1.14.0.