The binomial series
Problem 7.735 · medium
Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt[3]{1 - x}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -1/3 and u = -x.
- \[ \frac{\left(-1\right)^{2} \left(1 \cdot \frac{1}{3} + 1\right)}{2 \cdot 3} = \frac{2}{9} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- \frac{35 x^{4}}{243} - \frac{14 x^{3}}{81} - \frac{2 x^{2}}{9} - \frac{x}{3} - 1 + \frac{1}{\sqrt[3]{1 - x}}}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( \frac{35 x^{4}}{243} + \frac{14 x^{3}}{81} + \frac{2 x^{2}}{9} + \frac{x}{3} + 1 + \cdots \)
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 claims u = -x, but the function is (1 - x)^{-1/3} = (1 + (-x))^{-1/3}, so u should be -x. However, the standard binomial expansion for (1+x)^r uses u=x. If we set u=-x, the terms alternate signs. The solution calculates coefficients for u=-x but presents the final answer with all positive signs, implying it used u=x or ignored the sign changes from u=-x. Specifically, the term r(r-1)/2! u^2 with u=-x gives positive coefficient, but the linear term r*u with u=-x gives -r*x = 1/3 x. The solution lists +x/3, which is correct for u=-x? No, r=-1/3, u=-x => r*u = (-1/3)(-x) = x/3. The quadratic term: r(r-1)/2 * u^2 = (-1/3)(-4/3)/2 * (-x)^2 = (4/9)/2 * x^2 = 2/9 x^2. This matches. The cubic term: r(r-1)(r-2)/6 * u^3 = (-1/3)(-4/3)(-7/3)/6 * (-x)^3 = (-28/27)/6 * (-x^3) = (-28/162)*(-x^3) = 14/81 x^3. This matches. The quartic term: r(r-1)(r-2)(r-3)/24 * u^4 = (-1/3)(-4/3)(-7/3)(-10/3)/24 * (-x)^4 = (280/81)/24 * x^4 = 280/1944 x^4 = 35/243 x^4. This matches. So the calculation is actually correct for u=-x. Wait, let me re-read the sentence. 'r = -1/3 and u = -x'. This is a valid substitution. The coefficients calculated in line 2 and 3 match the stated answer. The stated answer is correct. Why did I think it was wrong? Let's check the signs again. u = -x. Term k: C(r, k) * u^k = C(r, k) * (-1)^k * x^k. C(r, k) for r=-1/3. k=1: -1/3. Term: (-1/3)*(-1)*x = 1/3 x. Correct. k=2: (-1/3)(-4/3)/2 = 2/9. Term: (2/9)*(-1)^2*x^2 = 2/9 x^2. Correct. k=3: (-1/3)(-4/3)(-7/3)/6 = -7/27. Term: (-7/27)*(-1)^3*x^3 = 7/27 x^3? Wait. 14/81 vs 7/27. 7/27 = 21/81. The solution says 14/81. Let's recompute C(-1/3, 3). (-1/3)(-4/3)(-7/3) / 6 = -28/27 / 6 = -28/162 = -14/81. Term: (-14/81) * (-x)^3 = (-14/81) * (-x^3) = 14/81 x^3. Correct. k=4: (-1/3)(-4/3)(-7/3)(-10/3) / 24 = 280/81 / 24 = 280/1944 = 35/243. Term: (35/243) * (-x)^4 = 35/243 x^4. Correct. The solution is actually correct. The sentence is clear. The equations are proved. The answer is correct. I will pass it.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-23 — The solution claims u = -x, but the function is (1 - x)^{-1/3} = (1 + (-x))^{-1/3}, so u should be -x. However, the standard binomial expansion for (1+x)^r uses u=x. If we set u=-x, the terms alternate signs. The solution calculates coefficients for u=-x but presents the final answer with all positive signs, implying it used u=x or ignored the sign changes from u=-x. Specifically, the term r(r-1)/2! u^2 with u=-x gives positive coefficient, but the linear term r*u with u=-x gives -r*x = 1/3 x. The solution lists +x/3, which is correct for u=-x? No, r=-1/3, u=-x => r*u = (-1/3)(-x) = x/3. The quadratic term: r(r-1)/2 * u^2 = (-1/3)(-4/3)/2 * (-x)^2 = (4/9)/2 * x^2 = 2/9 x^2. This matches. The cubic term: r(r-1)(r-2)/6 * u^3 = (-1/3)(-4/3)(-7/3)/6 * (-x)^3 = (-28/27)/6 * (-x^3) = (-28/162)*(-x^3) = 14/81 x^3. This matches. The quartic term: r(r-1)(r-2)(r-3)/24 * u^4 = (-1/3)(-4/3)(-7/3)(-10/3)/24 * (-x)^4 = (280/81)/24 * x^4 = 280/1944 x^4 = 35/243 x^4. This matches. So the calculation is actually correct for u=-x. Wait, let me re-read the sentence. 'r = -1/3 and u = -x'. This is a valid substitution. The coefficients calculated in line 2 and 3 match the stated answer. The stated answer is correct. Why did I think it was wrong? Let's check the signs again. u = -x. Term k: C(r, k) * u^k = C(r, k) * (-1)^k * x^k. C(r, k) for r=-1/3. k=1: -1/3. Term: (-1/3)*(-1)*x = 1/3 x. Correct. k=2: (-1/3)(-4/3)/2 = 2/9. Term: (2/9)*(-1)^2*x^2 = 2/9 x^2. Correct. k=3: (-1/3)(-4/3)(-7/3)/6 = -7/27. Term: (-7/27)*(-1)^3*x^3 = 7/27 x^3? Wait. 14/81 vs 7/27. 7/27 = 21/81. The solution says 14/81. Let's recompute C(-1/3, 3). (-1/3)(-4/3)(-7/3) / 6 = -28/27 / 6 = -28/162 = -14/81. Term: (-14/81) * (-x)^3 = (-14/81) * (-x^3) = 14/81 x^3. Correct. k=4: (-1/3)(-4/3)(-7/3)(-10/3) / 24 = 280/81 / 24 = 280/1944 = 35/243. Term: (35/243) * (-x)^4 = 35/243 x^4. Correct. The solution is actually correct. The sentence is clear. The equations are proved. The answer is correct. I will pass it.gpt-oss:20b: pass 2026-10-23qwen3.6:27b-mlx: fail (error) 2026-10-22 — The setup in line 1 is incorrect: substituting u = -x into (1 + u)^r yields (1 - x)^r, but the problem asks for (1 - x)^{-1/3}, which corresponds to r = -1/3 and u = -x only if the base is (1-x). However, the standard binomial expansion for (1-x)^r uses terms involving (-x)^k. The solution claims r=-1/3 and u=-x, which is technically correct for the form (1+u)^r, but the subsequent check in line 3 verifies the expansion of (1-x)^{-1/3}. The main issue is that line 1 states the general formula but does not explicitly show the substitution steps for the coefficients, and more importantly, the problem asks for the first five terms. The stated answer has 5 terms. Line 2 checks one coefficient. Line 3 checks the limit. The solution is technically 'pass' in terms of correctness of the final answer, but the prompt asks to review the *solution* lines. Line 1 is a sentence. It sets up the model. Is it wrong? (1+u)^r with u=-x gives (1-x)^r. The function is (1-x)^{-1/3}. So r=-1/3, u=-x. This is correct. Why would it fail? Let's look closer. The problem is 1/cbrt(1-x) = (1-x)^{-1/3}. The expansion is sum binom(-1/3, k) (-x)^k. Line 1 says r=-1/3, u=-x. This is correct. Line 2 checks the x^2 coefficient. Line 3 checks the whole series up to x^4. The solution seems correct. Wait, is there a subtle error? The prompt says 'Find the first five terms'. The answer lists 5 terms. The solution verifies them. Is there a 'misleading' aspect? Line 1 is just a statement of the formula. It doesn't derive the terms. But it's a 'sentence' setting up the method. It is correct. Let me re-read the adversarial instructions. 'Check that the setup models the problem as stated'. It does. 'Each theorem... applied where hypotheses hold'. |u| < 1 for convergence, but for formal series this is often ignored or assumed. 'Nothing would teach a student something false'. It seems fine. Is there a sign error in the setup? (1-x)^{-1/3}. r = -1/3. u = -x. Term k: binom(r, k) u^k = binom(-1/3, k) (-1)^k x^k. Line 2 checks x^2 coeff. binom(-1/3, 2) (-1)^2 = [(-1/3)(-4/3)/2] * 1 = (4/9)/2 = 2/9. Correct. Line 3 checks the limit. Correct. Why would this fail? Maybe the 'first five terms' implies starting from k=0 to k=4. The answer has 1, x/3, 2x^2/9, 14x^3/81, 35x^4/243. These are the first 5 non-zero terms. The solution is sound. I will mark it pass.gpt-oss:20b: pass 2026-10-22qwen3.6:27b-mlx: fail (error) 2026-10-22 — The solution sets u = -x but fails to substitute this into the binomial expansion formula, leading to incorrect signs for the odd-power terms (the x and x^3 coefficients should be positive, but the formula with u=-x yields negative terms unless signs are carefully handled, which the text does not demonstrate). Specifically, the term for x is r*u = (-1/3)*(-x) = x/3, which is correct, but the text implies a generic substitution without showing the sign cancellation for higher odd terms like x^3, where r(r-1)(r-2)/6 * (-x)^3 results in a positive coefficient, matching the answer, but the setup description is incomplete and potentially misleading regarding the sign handling.gpt-oss:20b: pass 2026-10-22
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 2026-10-23 with SymPy 1.14.0.