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The binomial series

Problem 7.588 · medium

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(1 - x\right)^{2}} \).

(1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -2 and u = -x.
Reviewed
\[ - \frac{2 \left(-1\right)^{2} \left(-2 - 1\right)}{2} = 3 \]
The x² coefficient, r(r − 1)/2 · c².✓ Proved
\[ \lim_{x \to 0}\left(\frac{- 5 x^{4} - 4 x^{3} - 3 x^{2} - 2 x - 1 + \frac{1}{\left(1 - x\right)^{2}}}{x^{4}}\right) = 0 \]
These terms agree with the function to order x⁴.✓ Proved

Answer \( 5 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 1 + \cdots \)

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	f minus the polynomial at x = 0.01 and 0.001

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the binomial expansion parameters and verifies the coefficients. Although the stated answer lists terms in descending order of power, this is mathematically equivalent to the standard ascending order and does not constitute an error.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the binomial expansion parameters and verifies the coefficients. Although the stated answer lists terms in descending order of power, this is mathematically equivalent to the standard ascending order and does not constitute an error.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The stated answer lists the terms in descending order of powers (5x^4 + ... + 1), which is non-standard for a series expansion and contradicts the ascending order implied by the binomial formula setup. Additionally, the solution fails to explicitly derive or list the first five terms as requested, instead only verifying the x^2 coefficient and the limit behavior.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution sets up the binomial expansion correctly but fails to actually derive or list the first five terms. It only verifies the coefficient of x^2 and checks the limit for the sum up to x^4, omitting the derivation of the coefficients for x, x^3, and x^4, and failing to state the final series explicitly as requested.
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/binomial_series, checked 2026-10-17 with SymPy 1.14.0.