The binomial series
Problem 7.637 · medium
Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(\frac{x}{2} + 1\right)^{2}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -2 and u = x/2.
- \[ \left(1 \cdot \frac{1}{2}\right)^{2} \left(-2\right) \left(-2 - 1\right) \frac{1}{2} = \frac{3}{4} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- \frac{5 x^{4}}{16} + \frac{x^{3}}{2} - \frac{3 x^{2}}{4} + x - 1 + \frac{1}{\left(\frac{x}{2} + 1\right)^{2}}}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( \frac{5 x^{4}}{16} - \frac{x^{3}}{2} + \frac{3 x^{2}}{4} - x + 1 + \cdots \)
Lines: 2 proved, 1 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 | — | 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: fail (style) — The solution states the binomial expansion formula and checks coefficients, but it never actually lists the first five terms of the series, so it does not fully answer the problem.qwen3.6:27b-mlx: fail (error) — The stated answer lists the terms in descending order of powers (x^4, x^3, ...), but the binomial series is a power series in ascending powers of x. The correct first five terms are 1 - x + 3/4 x^2 - 1/2 x^3 + 5/16 x^4. The solution's setup is correct, but the final answer format is non-standard and potentially confusing regarding the definition of 'first five terms' (which usually implies lowest powers first).
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The stated answer lists the terms in descending order of powers (x^4, x^3, ...), but the binomial series is a power series in ascending powers of x. The correct first five terms are 1 - x + 3/4 x^2 - 1/2 x^3 + 5/16 x^4. The solution's setup is correct, but the final answer format is non-standard and potentially confusing regarding the definition of 'first five terms' (which usually implies lowest powers first).gpt-oss:20b: fail (style) 2026-10-18 — The solution states the binomial expansion formula and checks coefficients, but it never actually lists the first five terms of the series, so it does not fully answer the problem.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The stated answer lists the terms in descending order of power (x^4 down to constant), which is not the standard form for a Maclaurin series expansion (which is ascending). Furthermore, the signs in the stated answer are incorrect for the standard expansion; the expansion of (1 + x/2)^{-2} is 1 - x + 3/4 x^2 - 1/2 x^3 + 5/16 x^4, whereas the stated answer has alternating signs starting with positive for the highest power, implying a sign error or a misunderstanding of the series structure.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer has incorrect signs for all terms except the constant term. The binomial expansion of (1 + x/2)^{-2} yields alternating signs stargpt-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 2026-10-18 with SymPy 1.14.0.