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Series convergence tests

Problem 7.230 · easy

Does \( \displaystyle \sum_{n=1}^{\infty} 5^{- n} n^{2} \) converge or diverge?

Use the ratio test.
Reviewed
\[ \frac{5^{n} 5^{- n - 1} \left(n + 1\right)^{2}}{n^{2}} = \frac{\left(n + 1\right)^{2}}{5 n^{2}} \]
a_(n+1)/a_n.✓ Proved
\[ \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{2}}{5 n^{2}}\right) = \frac{1}{5} \]
The limit is 1/5, < 1: converges.✓ Proved

Answer \( \text{converges} \)

Lines: 2 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	SymPy's own convergence decision (Sum.is_convergent) agrees

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the ratio test, computes the limit accurately, and draws the correct conclusion based on the limit being less than 1.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the ratio test, computes the limit accurately, and draws the correct conclusion based on the limit being less than 1.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The algebraic simplification in step 2 is incorrect. The term 5^n * 5^(-n-1) simplifies to 5^(-1) = 1/5, but the solution incorrectly writes the denominator as 5*n^2 while keeping the numerator as (n+1)^2, implying the factor of 5 was handled incorrectly or the expression for a_(n+1)/a_n was set up wrong (it should be 5^(-n-1)(n+1)^2 / (5^-n n^2) = (1/5) * ((n+1)/n)^2). Although the final limit 1/5 is correct, the intermediate equation shown is mathematically false.
gpt-oss:20b: pass 2026-10-03

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/series_convergence, checked 2026-10-03 with SymPy 1.14.0.