The binomial series
Problem 7.587 · medium
Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(1 - x\right)^{3}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -3 and u = -x.Reviewed
- \[ - \frac{3 \left(-1\right)^{2} \left(-3 - 1\right)}{2} = 6 \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- 15 x^{4} - 10 x^{3} - 6 x^{2} - 3 x - 1 + \frac{1}{\left(1 - x\right)^{3}}}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( 15 x^{4} + 10 x^{3} + 6 x^{2} + 3 x + 1 + \cdots \)
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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the binomial series formula and parameters. The algebraic checks confirm the coefficients match the stated answer, which is presented in descending order of powers but is mathematically equivalent.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the binomial series formula and parameters. The algebraic checks confirm the coefficients match the stated answer, which is presented in descending order of powers but is mathematically equivalent.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution sets up the binomial series with r = -3 and u = -x, but the stated answer has all positive coefficients, which corresponds to the expansion of (1 - x)^{-3} using the standard formula (1+x)^n with x replaced by -x, but the signs in the binomial coefficients for negative r alternate. Specifically, the term for x^1 is r*u = (-3)*(-x) = 3x, x^2 is (-3)(-4)/2 * (-x)^2 = 6x^2, x^3 is (-3)(-4)(-5)/6 * (-x)^3 = -10 * (-x^3) = 10x^3, x^4 is (-3)(-4)(-5)(-6)/24 * (-x)^4 = 15 * x^4. The coefficients are indeed 1, 3, 6, 10, 15. The stated answer lists them in descending order of power (15x^4 + ... + 1), which is unconventional but mathematically correct as a polynomial. However, the prompt asks for the 'first five terms', which usually implies ascending order of powers (1 + 3x + 6x^2 + 10x^3 + 15x^4). The provided answer is just a reordering. Is there an error? Let's re-read carefully. The solution says 'r = -3 and u = -x'. The expansion is sum C(r, k) u^k. C(-3, 1) = -3. u^1 = -x. Term = 3x. C(-3, 2) = 6. u^2 = x^2. Term = 6x^2. C(-3, 3) = -10. u^3 = -x^3. Term = 10x^3. C(-3, 4) = 15. u^4 = x^4. Term = 15x^4. The coefficients are correct. The order is reversed. Is reversing the order an 'error'? Usually, 'first five terms' implies the lowest powers first. Presenting them in descending order is misleading or at least non-standard for a series expansion starting from n=0. More importantly, the solution text only shows the setup and two checks. It does not explicitly derive the terms 1, 3x, 6x^2, 10x^3, 15x^4 in the text, it just jumps to a check. But the main issue is likely the ordering or the lack of explicit derivation steps for the terms themselves in the 'sentence' part, relying on the equation check. However, the biggest red flag is that the solution doesn't actually *state* the terms in the sentences, it just checks them. But the prompt asks to review the solution. The solution provided is very sparse. It sets up the formula, checks one coefficient, and checks the limit. It never explicitly lists the terms 1, 3x, 6x^2, 10x^3, 15x^4 in the text. The 'Stated answer' is separate. The solution itself is incomplete as a derivation. But is it 'wrong'? The check `Limit(...) = 0` confirms the polynomial `15x^4 + ... + 1` matches the function. The order is just a presentation choice. Wait, look at the stated answer: `15 x^{4} + 10 x^{3} + 6 x^{2} + 3 x + 1`. This is correct. The solution is technically correct but poor style. However, often 'first five terms' implies the Maclaurin series terms $a_0, a_1, a_2, a_3, a_4$. Listing them in reverse is confusing. I will mark it as misleading because it presents the series in descending order of powers, which is contrary to the standard convention for 'first N terms' of a power series (which are the lowest degree terms).gpt-oss:20b: fail (error) 2026-10-15 — The solution omits the required first five terms of the series; it only verifies a single coefficient and a limit check, so it does not answer the problem as stated.qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution sets up the binomial expansion with r = -3 and u = -x, but the stated answer lists terms with positive coefficients (1, 3, 6, 10, 15),gpt-oss:20b: fail (style) 2026-10-14 — The solution correctly sets up the series and verifies coefficients, but it never explicitly lists the first five terms requested in the problem statement, leaving the conclusion incomplete.
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.