The binomial series
Problem 7.632 · medium
Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(\frac{x}{2} + 1\right)^{3}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -3 and u = x/2.
- \[ \left(1 \cdot \frac{1}{2}\right)^{2} \left(-3\right) \left(-3 - 1\right) \frac{1}{2} = \frac{3}{2} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- \frac{15 x^{4}}{16} + \frac{5 x^{3}}{4} - \frac{3 x^{2}}{2} + \frac{3 x}{2} - 1 + \frac{1}{\left(\frac{x}{2} + 1\right)^{3}}}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( \frac{15 x^{4}}{16} - \frac{5 x^{3}}{4} + \frac{3 x^{2}}{2} - \frac{3 x}{2} + 1 + \cdots \)
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 does not explicitly list the first five terms of the series; it only verifies the coefficient of x² and uses a limit to check agreement up to x⁴, leaving the required terms unshown.qwen3.6:27b-mlx: fail (error) — The stated answer has incorrect signs for all terms except the constant. The binomial expansion of (1 + x/2)^(-3) yields alternating signs starting with a positive linear term (+3x/2), but the answer lists -3x/2. The solution setup correctly identifies r=-3 and u=x/2, but the final result provided in the problem statement is wrong, and the solution fails to derive the correct signs, instead just checking an incorrect polynomial against the function (which would fail if checked properly, but the CAS check in line 3 is for the negative of the correct series, implying the solution is validating the wrong answer).
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The stated answer has incorrect signs for all terms except the constant. The binomial expansion of (1 + x/2)^(-3) yields alternating signs starting with a positive linear term (+3x/2), but the answer lists -3x/2. The solution setup correctly identifies r=-3 and u=x/2, but the final result provided in the problem statement is wrong, and the solution fails to derive the correct signs, instead just checking an incorrect polynomial against the function (which would fail if checked properly, but the CAS check in line 3 is for the negative of the correct series, implying the solution is validating the wrong answer).gpt-oss:20b: fail (style) 2026-10-18 — The solution does not explicitly list the first five terms of the series; it only verifies the coefficient of x² and uses a limit to check agreement up to x⁴, leaving the required terms unshown.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The stated answer has incorrect signs for all terms except the constant term. The binomial expansion of (1 + x/2)^(-3) yields alternating signs starting with positive for the constant, negative for x, positive for x^2, etc. The provided answer has -3x/2 (correct sign) but +3x^2/2 (correct sign) and -5x^3/4 (correct sign) and +15x^4/16 (correct sign). Wait, let me re-evaluate. Binomial series for (1+u)^r with r=-3, u=x/2: Term 0: 1 Term 1: -3(x/2) = -3x/2 Term 2: (-3)(-4)/2 * (x/2)^2 = 6 * x^2/4 = 3x^2/2 Term 3: (-3)(-4)(-5)/6 * (x/2)^3 = -10 * x^3/8 = -5x^3/4 Term 4: (-3)(-4)(-5)(-6)/24 * (x/2)^4 = 15 * x^4/16 = 15x^4/16 So the correct series is 1 - 3x/2 + 3x^2/2 - 5x^3/4 + 15x^4/16. The stated answer is: 15x^4/16 - 5x^3/4 + 3x^2/2 - 3x/2 + 1. This is exactly the correct series, just written in descending order of powers. Let's re-read the solution steps. Step 1 sets up the formula correctly. Step 2 checks the x^2 coefficient. It calculates 3/2. This matches the term 3x^2/2 in the stated answer. Step 3 checks the limit of the difference between the function and the polynomial (15x^4/16 - 5x^3/4 + 3x^2/2 + 3x/2 - 1) divided by x^4. Wait, look at Step 3's polynomial: `(-15*x**4/16 + 5*x**3/4 - 3*x**2/2 + 3*x/2 - 1)`. The signs in the limit check are: -15/16, +5/4, -3/2, +3/2, -1. The stated answer has: +15/16, -5/4, +3/2, -3/2, +1. The polynomial in the limit check is the NEGATIVE of the stated answer (except for the constant term sign? No, -1 vs +1). Actually, the limit checks if `Function - Polynomial` is o(x^4). If the Polynomial is `P`, and `P` is the negative of the correct series `S`, then `Function - P = S - (-S) = 2S`. The limit of `2S/x^4` as x->0 is not 0 (it's 2 * 15/16 = 15/8). However, the equation says the limit IS 0. Let's look closer at the polynomial in step 3: `-15*x**4/16 + 5*x**3/4 - 3*x**2/2 + 3*x/2 - 1`. This polynomial is `- (15*x**4/16 - 5*x**3/4 + 3*x**2/2 - 3*x/2 + 1)`. This is `- (Stated Answer)`. So Step 3 claims that `(Function) - (- Stated Answer)` is o(x^4). This implies `Function + Stated Answer` is o(x^4). But `Function` is approximately `Stated Answer`. So `2 * Stated Answer` is o(x^4), which is false. Wait, did I misread the signs in Step 3? `Limit((-15*x**4/16 + 5*x**3/4 - 3*x**2/2 + 3*x/2 - 1 + (x/2 + 1)**(-3))/x**4, ...)` Let `S` be the correct series `1 - 3x/2 + 3x^2/2 - 5x^3/4 + 15x^4/16`. The polynomial in the limit is `P = -15x^4/16 + 5x^3/4 - 3x^2/2 + 3x/2 - 1`. Note that `P = -S`. The limit checks `(f(x) + S(x)) / x^4`. Since `f(x) ~ S(x)`, this limit is `2 * 15/16 != 0`. So the equation in Step 3 is FALSE. The limit is not 0. Therefore, the verification step is mathematically incorrect. It claims a limit is 0 when it is not. This is an error in the solution's verification logic.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 the terms (it alternates starting with positive for x^4, but the expansion of (1+x/2)^{-3} has altergpt-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.