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Partial sums of a series

Problem 7.561 · easy

Find the first four partial sums \( \displaystyle S_1, \dots, S_4 \) of \( \displaystyle \sum_{n=1}^{\infty} \frac{1}{n \left(n + 1\right)} \).

S_k adds the first k terms: a₁ = 1/2, a₂ = 1/6, and so on.
Reviewed
\[ \left. 1 \frac{1}{n \left(n + 1\right)} \right|_{\substack{ n=1 }} = \frac{1}{2} \]
S1.✓ Proved
\[ 1 \cdot \frac{1}{6} + 1 \cdot \frac{1}{2} = \frac{2}{3} \]
S2.✓ Proved
\[ 1 \cdot \frac{1}{12} + 1 \cdot \frac{1}{6} + 1 \cdot \frac{1}{2} = \frac{3}{4} \]
S3.✓ Proved
\[ 1 \cdot \frac{1}{20} + 1 \cdot \frac{1}{12} + 1 \cdot \frac{1}{6} + 1 \cdot \frac{1}{2} = \frac{4}{5} \]
S4.✓ Proved

Answer \( S_1 = \frac{1}{2},\ S_2 = \frac{2}{3},\ S_3 = \frac{3}{4},\ S_4 = \frac{4}{5} \)

✓ Nihil obstat Lines: 4 proved, 1 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	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
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
5	✓ 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	the terms added in floating point

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the terms and computes the partial sums. The arithmetic is verified by the CAS, and the setup is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the terms and computes the partial sums. The arithmetic is verified by the CAS, and the setup is correct.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the terms and computes the partial sums. The algebraic steps are verified, and the final results match the stated answer.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the terms and computes the partial sums. The arithmetic is correct, and the setup matches the problem statement.
gpt-oss:20b: pass 2026-10-14

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/partial_sums, checked 2026-10-17 with SymPy 1.14.0.