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Geometric series

Problem 7.9 · medium Video

Does \( \displaystyle \sum_{n=0}^{\infty} - 5^{- n} \) converge? If so, find its sum.

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

Answer \( - \frac{5}{4} \)

Bars show the terms -1/5^n for n = 0 to 23; dots show the running totals. The totals level off at -5/4.

✓ 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, applies the geometric series convergence test and sum formula correctly, and arrives at the correct result.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the first term and common ratio, applies the geometric series convergence test and sum formula correctly, and arrives at the correct result.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the first term and common ratio of the geometric series, applies the convergence test and sum formula correctly, and arrives at the correct result.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26
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.