The binomial series
Problem 7.667 · medium
Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(3 x + 1\right)^{3}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -3 and u = 3*x.
- \[ - \frac{3 \cdot 3^{2} \left(-3 - 1\right)}{2} = 54 \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- 1215 x^{4} + 270 x^{3} - 54 x^{2} + 9 x - 1 + \frac{1}{\left(3 x + 1\right)^{3}}}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( 1215 x^{4} - 270 x^{3} + 54 x^{2} - 9 x + 1 + \cdots \)
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: passqwen3.6:27b-mlx: fail (error) — The stated answer lists the terms in descending order of power (1215x^4 ... + 1), which is not the standard form for a Maclaurin series (ascending powers). Furthermore, the signs in the stated answer are incorrect for the expansion of (1+3x)^{-3}; the correct expansion is 1 - 9x + 54x^2 - 270x^3 + 1215x^4. The stated answer has alternating signs starting with positive for the highest power, but the coefficients' signs relative to each other are flipped compared to the standard series expansion (e.g., the x term should be -9x, not -9x if written as ... - 9x + 1, but the stated answer writes -9x + 1, which is correct for that term, but the x^3 term is -270x^3, which is correct, and x^4 is 1215x^4, which is correct. Wait, let's re-read carefully. Stated: 1215 x^4 - 270 x^3 + 54 x^2 - 9 x + 1. This is actually the correct polynomial, just written in reverse order. However, the solution text says 'Find the first five terms...'. Usually, this implies ascending order. More critically, the solution only verifies the x^2 coefficient and the limit for the whole polynomial. It does not derive the other coefficients. But the main issue is likely the presentation or a subtle sign error? Let's check the signs. (1+u)^-3 = 1 - 3u + 6u^2 - 10u^3 + 15u^4. u=3x. 1 - 9x + 54x^2 - 270x^3 + 1215x^4. The stated answer is 1215 x^4 - 270 x^3 + 54 x^2 - 9 x + 1. This is mathematically equivalent. Is it 'wrong'? No. Is it 'misleading'? Maybe. But look at the solution steps. Step 1 sets up the formula. Step 2 checks one coefficient. Step 3 checks the limit. The solution is extremely sparse. It doesn't actually *find* the terms, it just verifies a given answer. The prompt asks to 'Find' the terms. The solution provided is a verification, not a derivation. This is a methodological failure in the context of 'Find'. Also, the stated answer order is non-standard. But the biggest error is that the solution does not actually perform the calculation for all terms, it just asserts the answer and checks it partially. A student would not learn how to find the terms. It's a 'pass' on correctness of the final expression, but 'fail' on the solution process being a mere verification rather than a derivation. However, usually, these reviews check for mathematical errors. Let's look closer. Is there an error? No mathematical error in the final expression. But the solution text is just two checks. It fails to 'Find' the terms. It's a 'misleading' or 'error' in that it doesn't solve the problem as stated. Let's call it 'error' because the solution does not derive the result.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The stated answer lists the terms in descending order of power (1215x^4 ... + 1), which is not the standard form for a Maclaurin series (ascending powers). Furthermore, the signs in the stated answer are incorrect for the expansion of (1+3x)^{-3}; the correct expansion is 1 - 9x + 54x^2 - 270x^3 + 1215x^4. The stated answer has alternating signs starting with positive for the highest power, but the coefficients' signs relative to each other are flipped compared to the standard series expansion (e.g., the x term should be -9x, not -9x if written as ... - 9x + 1, but the stated answer writes -9x + 1, which is correct for that term, but the x^3 term is -270x^3, which is correct, and x^4 is 1215x^4, which is correct. Wait, let's re-read carefully. Stated: 1215 x^4 - 270 x^3 + 54 x^2 - 9 x + 1. This is actually the correct polynomial, just written in reverse order. However, the solution text says 'Find the first five terms...'. Usually, this implies ascending order. More critically, the solution only verifies the x^2 coefficient and the limit for the whole polynomial. It does not derive the other coefficients. But the main issue is likely the presentation or a subtle sign error? Let's check the signs. (1+u)^-3 = 1 - 3u + 6u^2 - 10u^3 + 15u^4. u=3x. 1 - 9x + 54x^2 - 270x^3 + 1215x^4. The stated answer is 1215 x^4 - 270 x^3 + 54 x^2 - 9 x + 1. This is mathematically equivalent. Is it 'wrong'? No. Is it 'misleading'? Maybe. But look at the solution steps. Step 1 sets up the formula. Step 2 checks one coefficient. Step 3 checks the limit. The solution is extremely sparse. It doesn't actually *find* the terms, it just verifies a given answer. The prompt asks to 'Find' the terms. The solution provided is a verification, not a derivation. This is a methodological failure in the context of 'Find'. Also, the stated answer order is non-standard. But the biggest error is that the solution does not actually perform the calculation for all terms, it just asserts the answer and checks it partially. A student would not learn how to find the terms. It's a 'pass' on correctness of the final expression, but 'fail' on the solution process being a mere verification rather than a derivation. However, usually, these reviews check for mathematical errors. Let's look closer. Is there an error? No mathematical error in the final expression. But the solution text is just two checks. It fails to 'Find' the terms. It's a 'misleading' or 'error' in that it doesn't solve the problem as stated. Let's call it 'error' because the solution does not derive the result.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer has the wrong signs for all terms except the constant. The expansion of (1+3x)^{-3} is 1 - 9x + 54x^2 - 270x^3 + 1215x^4, but thegpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer lists the terms in descending order of powers (x^4 down to x^0), which is non-standard for a Maclaurin/binomial series expansion.gpt-oss:20b: fail (error) 2026-10-15 — The solution only verifies the x² coefficient and checks agreement up to order x⁴, but it never actually lists the first five terms of the series. The conclusion that “these terms agree with the function to order x⁴” is unsupported because the other coefficients are not shown or derived.
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.