The binomial series
Problem 7.638 · medium
Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(1 - 2 x\right)^{3}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -3 and u = -2*x.
- \[ - \frac{3 \left(-2\right)^{2} \left(-3 - 1\right)}{2} = 24 \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- 240 x^{4} - 80 x^{3} - 24 x^{2} - 6 x - 1 + \frac{1}{\left(1 - 2 x\right)^{3}}}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( 240 x^{4} + 80 x^{3} + 24 x^{2} + 6 x + 1 + \cdots \)
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 solution sets u = -2x but the binomial series formula requires (1+u). Since 1-2x = 1+(-2x), u should be -2x. However, the stated answer has positive coefficients, which corresponds to expanding (1-2x)^{-3} directly or using u=2x with r=-3? No, (1-2x)^{-3} = 1 + (-3)(-2x) + ... = 1 + 6x + ... The coefficients are positive. The setup says u = -2x. Let's check the first term: 1 + r*u = 1 + (-3)*(-2x) = 1 + 6x. This matches. The second term: r(r-1)/2 * u^2 = (-3)(-4)/2 * (-2x)^2 = 6 * 4x^2 = 24x^2. This matches. The setup is actually correct. Wait, let me re-read carefully. The problem asks for the first five terms. The stated answer lists 5 terms. The solution only verifies the x^2 coefficient and the limit for the whole polynomial. It does not explicitly calculate the x^3 and x^4 coefficients in the text, though the limit check implies they are correct. Is there an error? The sentence says 'u = -2*x'. The formula is for (1+u)^r. Here we have (1-2x)^{-3} = (1+(-2x))^{-3}. So u=-2x is correct. The coefficients calculated are correct. The limit check confirms the polynomial matches the function up to order 4. The solution seems mathematically sound, but it skips the explicit derivation of the x^3 and x^4 terms in the text, relying on the limit check. Is this an error? The prompt asks to review the solution. The solution provides the formula, one coefficient check, and a global limit check. It does not explicitly state the x^3 and x^4 coefficients in the text, but the final answer is given. The 'Stated answer' is separate from the 'Solution' lines? No, the 'Stated answer' is the target. The 'Solution' is the work. The work does not show the calculation for x^3 and x^4. It only shows x^2. Then it checks the limit. This is a valid verification method. However, usually 'Find the terms' requires showing the terms. The solution is a bit sparse but not wrong. Let me look closer. Is there a sign error? r=-3, u=-2x. Term 1: 1. Term 2: -3(-2x) = 6x. Term 3: 6(4x^2) = 24x^2. Term 4: (-3)(-4)(-5)/6 * (-2x)^3 = -10 * (-8x^3) = 80x^3. Term 5: (-3)(-4)(-5)(-6)/24 * (-2x)^4 = 15 * 16x^4 = 240x^4. The coefficients are 1, 6, 24, 80, 240. The stated answer is 240x^4 + 80x^3 + 24x^2 + 6x + 1. This is correct. The solution text is minimal but correct. Why would it fail? Maybe the sentence 'u = -2*x' is considered misleading if the student doesn't realize u is the entire term inside the parenthesis minus 1? No, that's standard. Let's look at the limit line. `Limit((-240*x**4 - 80*x**3 - 24*x**2 - 6*x - 1 + (1 - 2*x)**(-3))/x**4, x, 0, dir='+-') = 0`. This checks that the polynomial P(x) = 1 + 6x + 24x^2 + 80x^3 + 240x^4 approximates the function to order 4. This is a valid check. The solution is correct. Wait, is the order of terms in the stated answer an issue? It's descending. The problem asks for 'first five terms', usually implying ascending powers. But the set of terms is correct. The solution doesn't explicitly derive the higher terms, just verifies them. This might be considered 'misleading' or 'style' if it doesn't show the work, but it's not an error. However, looking at the strictness of 'adversarial review', skipping the derivation of 3 out of 5 terms might be seen as incomplete. But the limit check is a valid mathematical proof that those are the correct coefficients. I will mark it as pass because the logic is sound, even if brief. Actually, let me double check the limit expression. It subtracts the polynomial from the function. If the limit is 0, the polynomial is the Taylor polynomial. This is correct. I will pass it.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution sets u = -2x but the binomial series formula requires (1+u). Since 1-2x = 1+(-2x), u should be -2x. However, the stated answer has positive coefficients, which corresponds to expanding (1-2x)^{-3} directly or using u=2x with r=-3? No, (1-2x)^{-3} = 1 + (-3)(-2x) + ... = 1 + 6x + ... The coefficients are positive. The setup says u = -2x. Let's check the first term: 1 + r*u = 1 + (-3)*(-2x) = 1 + 6x. This matches. The second term: r(r-1)/2 * u^2 = (-3)(-4)/2 * (-2x)^2 = 6 * 4x^2 = 24x^2. This matches. The setup is actually correct. Wait, let me re-read carefully. The problem asks for the first five terms. The stated answer lists 5 terms. The solution only verifies the x^2 coefficient and the limit for the whole polynomial. It does not explicitly calculate the x^3 and x^4 coefficients in the text, though the limit check implies they are correct. Is there an error? The sentence says 'u = -2*x'. The formula is for (1+u)^r. Here we have (1-2x)^{-3} = (1+(-2x))^{-3}. So u=-2x is correct. The coefficients calculated are correct. The limit check confirms the polynomial matches the function up to order 4. The solution seems mathematically sound, but it skips the explicit derivation of the x^3 and x^4 terms in the text, relying on the limit check. Is this an error? The prompt asks to review the solution. The solution provides the formula, one coefficient check, and a global limit check. It does not explicitly state the x^3 and x^4 coefficients in the text, but the final answer is given. The 'Stated answer' is separate from the 'Solution' lines? No, the 'Stated answer' is the target. The 'Solution' is the work. The work does not show the calculation for x^3 and x^4. It only shows x^2. Then it checks the limit. This is a valid verification method. However, usually 'Find the terms' requires showing the terms. The solution is a bit sparse but not wrong. Let me look closer. Is there a sign error? r=-3, u=-2x. Term 1: 1. Term 2: -3(-2x) = 6x. Term 3: 6(4x^2) = 24x^2. Term 4: (-3)(-4)(-5)/6 * (-2x)^3 = -10 * (-8x^3) = 80x^3. Term 5: (-3)(-4)(-5)(-6)/24 * (-2x)^4 = 15 * 16x^4 = 240x^4. The coefficients are 1, 6, 24, 80, 240. The stated answer is 240x^4 + 80x^3 + 24x^2 + 6x + 1. This is correct. The solution text is minimal but correct. Why would it fail? Maybe the sentence 'u = -2*x' is considered misleading if the student doesn't realize u is the entire term inside the parenthesis minus 1? No, that's standard. Let's look at the limit line. `Limit((-240*x**4 - 80*x**3 - 24*x**2 - 6*x - 1 + (1 - 2*x)**(-3))/x**4, x, 0, dir='+-') = 0`. This checks that the polynomial P(x) = 1 + 6x + 24x^2 + 80x^3 + 240x^4 approximates the function to order 4. This is a valid check. The solution is correct. Wait, is the order of terms in the stated answer an issue? It's descending. The problem asks for 'first five terms', usually implying ascending powers. But the set of terms is correct. The solution doesn't explicitly derive the higher terms, just verifies them. This might be considered 'misleading' or 'style' if it doesn't show the work, but it's not an error. However, looking at the strictness of 'adversarial review', skipping the derivation of 3 out of 5 terms might be seen as incomplete. But the limit check is a valid mathematical proof that those are the correct coefficients. I will mark it as pass because the logic is sound, even if brief. Actually, let me double check the limit expression. It subtracts the polynomial from the function. If the limit is 0, the polynomial is the Taylor polynomial. This is correct. I will pass it.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution sets up the binomial series formula but fails to explicitly calculate or state the first five terms (coefficients for x^0 through x^4) in the text, relying instead on a limit check for the final answer. A complete solution should derive the coefficients 1, 6, 24, 80, and 240 step-by-step.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 solution sets u = -2x, but the binomial expansion of (1 - 2x)^{-3} requires u = -2x only if the base is (1+u). Here the base is (1-2x), so u shogpt-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.