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Series convergence tests

Problem 7.205 · easy Video

Does \( \displaystyle \sum_{n=1}^{\infty} 3^{- n} n \) converge or diverge?

Use the ratio test.
Reviewed
\[ \frac{3^{n} 3^{- n - 1} \left(n + 1\right)}{n} = \frac{n + 1}{3 n} \]
a_(n+1)/a_n.✓ Proved
\[ \lim_{n \to \infty}\left(\frac{n + 1}{3 n}\right) = \frac{1}{3} \]
The limit is 1/3, < 1: converges.✓ Proved

Answer \( \text{converges} \)

Bars show the terms n/3^n for n = 1 to 24; dots show the running totals. The totals level off: the series 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. How we verify

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

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-04
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies the ratio test. The limit calculation is correct, and the conclusion that the series converges because the limit is less than 1 is valid.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: fail (error) 2026-09-29 — The ratio calculation in step 2 is incorrect; the ratio a_{n+1}/a_n simplifies to (n+1)/(3n), not 3^n * 3^{-n-1} * (n+1)/n. While the final limit 1/3 is correct, the intermediate algebraic expression is wrong.
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/series_convergence, checked 2026-10-04 with SymPy 1.14.0.