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Geometric series

Problem 7.125 · medium Video

Does \( \displaystyle \sum_{n=1}^{\infty} 4 \left(- \frac{2}{3}\right)^{n - 1} \) converge? If so, find its sum.

\[ 4 \]
The first term, a.Reviewed
\[ - \frac{2}{3} \]
The common ratio, r.Reviewed
|r| = 2/3 < 1, so the series converges to a/(1 − r).
Reviewed
\[ \frac{12}{5} \]
a/(1 − r).Reviewed

Answer \( \frac{12}{5} \)

Bars show the terms 4(-2/3)^(n - 1) for n = 1 to 24; dots show the running totals. The totals level off at 12/5.

✓ Nihil obstat Lines: 4 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
2	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
3	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
4	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the 400th partial sum agrees

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the first term and common ratio, verifies the convergence condition for a geometric series, and applies the sum formula correctly.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the first term and common ratio, verifies the convergence condition for a geometric series, and applies the sum formula correctly.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the series as geometric, verifies the convergence condition |r| < 1, and applies the correct sum formula a/(1-r) with the correct first term and ratio.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the first term and common ratio, applies the geometric series convergence test and sum formula correctly, and arrives at the right answer.
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/geometric_series, checked 2026-10-13 with SymPy 1.14.0.