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Geometric series

Problem 7.201 · easy Video

Does \( \displaystyle \sum_{n=1}^{\infty} 5 \left(\frac{4}{3}\right)^{n - 1} \) converge? If so, find its sum.

\[ 5 \]
The first term, a.
\[ \frac{4}{3} \]
The common ratio, r.
|r| = 4/3 ≥ 1, so the series diverges.

Answer \( \text{diverges} \)

Bars show the terms 5(4/3)^(n - 1) for n = 1 to 24; dots show the running totals. The totals never settle: the series diverges.

Lines: 3 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	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
2	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
3	Not checked	—	a sentence; read, not computed
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the terms do not shrink (60th term checked)

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The divergence test for geometric series requires |r| > 1, not |r| >= 1. If |r| = 1, the series diverges by the nth term test, but the condition stated is mathematically imprecise and misleading regarding the standard geometric series theorem.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The divergence test for geometric series requires |r| > 1, not |r| >= 1. If |r| = 1, the series diverges by the nth term test, but the condition stated is mathematically imprecise and misleading regarding the standard geometric series theorem.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies the series as geometric with |r| > 1 and correctly concludes divergence.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies the series as geometric with ratio r=4/3. Since |r| > 1, the series diverges. The logic is sound and the conclusion is correct.
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/geometric_series, checked 2026-10-13 with SymPy 1.14.0.