The binomial series
Problem 7.597 · medium
Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt[3]{1 - 2 x}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -1/3 and u = -2*x.
- \[ \frac{\left(-2\right)^{2} \left(1 \cdot \frac{1}{3} + 1\right)}{2 \cdot 3} = \frac{8}{9} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- \frac{560 x^{4}}{243} - \frac{112 x^{3}}{81} - \frac{8 x^{2}}{9} - \frac{2 x}{3} - 1 + \frac{1}{\sqrt[3]{1 - 2 x}}}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( \frac{560 x^{4}}{243} + \frac{112 x^{3}}{81} + \frac{8 x^{2}}{9} + \frac{2 x}{3} + 1 + \cdots \)
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 setup in line 1 is incorrect: it states u = -2x, but the function is (1 - 2x)^(-1/3), which corresponds to u = -2x only if the base is (1+u). However, the standard binomial expansion for (1+x)^r uses x directly. Here, substituting u = -2x into (1+u)^r gives (1-2x)^r, which is correct. Wait, let's re-read carefully. The problem is 1/(1-2x)^(1/3) = (1-2x)^(-1/3). The formula is (1+u)^r. If we set u = -2x, then (1+(-2x))^(-1/3) = (1-2x)^(-1/3). This is correct. Let's check the coefficients. r = -1/3. Term 1: 1. Term 2: ru = (-1/3)(-2x) = 2/3 x. Correct. Term 3: r(r-1)/2 u^2 = (-1/3)(-4/3)/2 * (-2x)^2 = (4/9)/2 * 4x^2 = 2/9 * 4x^2 = 8/9 x^2. Correct. Term 4: r(r-1)(r-2)/6 u^3 = (-1/3)(-4/3)(-7/3)/6 * (-8x^3) = (-28/27)/6 * (-8x^3) = (-28/162) * (-8x^3) = (14/81) * 8x^3 = 112/81 x^3. Correct. Term 5: r(r-1)(r-2)(r-3)/24 u^4 = (-1/3)(-4/3)(-7/3)(-10/3)/24 * 16x^4 = (280/81)/24 * 16x^4 = (280/1944) * 16x^4 = (35/243) * 16x^4 = 560/243 x^4. Correct. The solution is actually correct. Why did I think it was wrong? Let me re-read the prompt's constraints. 'Find what is wrong'. Is there a subtle error? Line 1 says 'u = -2*x'. Line 2 calculates the x^2 coefficient. Line 3 checks the limit. The solution seems sound. However, usually, these adversarial reviews look for a specific flaw. Let's look at the notation. 'r = -1/3 and u = -2*x'. This is a valid substitution. The coefficients match. The limit check confirms the series. Is it possible the question implies finding the terms in a specific order or format? The stated answer lists them in descending order of power? No, ascending. 1, x, x^2, x^3, x^4. The stated answer is 560/243 x^4 + ... + 1. This is just a sum, order doesn't matter for equality. But typically 'first five terms' implies listing them. The solution provides the check. Is there a missing step? The solution jumps from the formula to a check of the x^2 term and then a global check. It doesn't explicitly list all terms in the text, but the 'Stated answer' is provided separately. The solution lines are just the derivation/check. Line 1 is the setup. Line 2 is a partial check. Line 3 is a full check. This is a valid verification method. Is there an error in the text of line 1? 'r(r-1)/2! u^2'. The factorial is 2!. Correct. 'r(r-1)(r-2)/3! u^3'. Correct. I will mark it as pass because it is mathematically sound.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The setup in line 1 is incorrect: it states u = -2x, but the function is (1 - 2x)^(-1/3), which corresponds to u = -2x only if the base is (1+u). However, the standard binomial expansion for (1+x)^r uses x directly. Here, substituting u = -2x into (1+u)^r gives (1-2x)^r, which is correct. Wait, let's re-read carefully. The problem is 1/(1-2x)^(1/3) = (1-2x)^(-1/3). The formula is (1+u)^r. If we set u = -2x, then (1+(-2x))^(-1/3) = (1-2x)^(-1/3). This is correct. Let's check the coefficients. r = -1/3. Term 1: 1. Term 2: ru = (-1/3)(-2x) = 2/3 x. Correct. Term 3: r(r-1)/2 u^2 = (-1/3)(-4/3)/2 * (-2x)^2 = (4/9)/2 * 4x^2 = 2/9 * 4x^2 = 8/9 x^2. Correct. Term 4: r(r-1)(r-2)/6 u^3 = (-1/3)(-4/3)(-7/3)/6 * (-8x^3) = (-28/27)/6 * (-8x^3) = (-28/162) * (-8x^3) = (14/81) * 8x^3 = 112/81 x^3. Correct. Term 5: r(r-1)(r-2)(r-3)/24 u^4 = (-1/3)(-4/3)(-7/3)(-10/3)/24 * 16x^4 = (280/81)/24 * 16x^4 = (280/1944) * 16x^4 = (35/243) * 16x^4 = 560/243 x^4. Correct. The solution is actually correct. Why did I think it was wrong? Let me re-read the prompt's constraints. 'Find what is wrong'. Is there a subtle error? Line 1 says 'u = -2*x'. Line 2 calculates the x^2 coefficient. Line 3 checks the limit. The solution seems sound. However, usually, these adversarial reviews look for a specific flaw. Let's look at the notation. 'r = -1/3 and u = -2*x'. This is a valid substitution. The coefficients match. The limit check confirms the series. Is it possible the question implies finding the terms in a specific order or format? The stated answer lists them in descending order of power? No, ascending. 1, x, x^2, x^3, x^4. The stated answer is 560/243 x^4 + ... + 1. This is just a sum, order doesn't matter for equality. But typically 'first five terms' implies listing them. The solution provides the check. Is there a missing step? The solution jumps from the formula to a check of the x^2 term and then a global check. It doesn't explicitly list all terms in the text, but the 'Stated answer' is provided separately. The solution lines are just the derivation/check. Line 1 is the setup. Line 2 is a partial check. Line 3 is a full check. This is a valid verification method. Is there an error in the text of line 1? 'r(r-1)/2! u^2'. The factorial is 2!. Correct. 'r(r-1)(r-2)/3! u^3'. Correct. I will mark it as pass because it is mathematically sound.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution sets u = -2x but the problem requires the expansion of (1 - 2x)^(-1/3), which corresponds to u = -2x in the standard form (1+u)^r. However, the stated answer has positive coefficients, implying the expansion of (1 - 2x)^(-1/3) where the terms are calculated as (-1)^k * binomial * (-2x)^k. Wait, let's look closer. The problem is 1/(1-2x)^(1/3) = (1-2x)^(-1/3). The binomial series for (1+u)^r with u=-2x gives terms involving (-2x)^k. The coefficient of x^1 is r*(-2) = (-1/3)*(-2) = 2/3. The coefficient of x^2 is r(r-1)/2 * (-2)^2 = (-1/3)(-4/3)/2 * 4 = (4/9)/2 * 4 = 2/9 * 4 = 8/9. The coefficient of x^3 is r(r-1)(r-2)/6 * (-2)^3 = (-1/3)(-4/3)(-7/3)/6 * (-8) = (-28/27)/6 * (-8) = (-28/162) * (-8) = 224/162 = 112/81. The coefficient of x^4 is r(r-1)(r-2)(r-3)/24 * (-2)^4 = (-1/3)(-4/3)(-7/3)(-10/3)/24 * 16 = (280/81)/24 * 16 = 280/1944 * 16 = 4480/1944 = 560/243. The stated answer matches these calculations. The sentence says u = -2*x. This is correct for the form (1+u)^r. The equation checks the x^2 coefficient. The limit check verifies the first 5 terms. The solution is actually correct. Why did I think it was wrong? Let me re-read the prompt's instruction. 'Find what is wrong'. Is there a subtle error? The sentence says 'r = -1/3 and u = -2*x'. The formula is for (1+u)^r. So (1 + (-2x))^(-1/3) = (1-2x)^(-1/3). This is correct. The coefficients in the stated answer are positive. Let's check the sign of the x^3 term in the series expansion. Term k=3: binom(-1/3, 3) * (-2x)^3. binom(-1/3, 3) = (-1/3)(-4/3)(-7/3)/6 = -28/162 = -14/81. (-2x)^3 = -8x^3. Product: (-14/81)*(-8) = 112/81. Positive. Term k=4: binom(-1/3, 4) * (-2x)^4. binom(-1/3, 4) = (-1/3)(-4/3)(-7/3)(-10/3)/24 = 280/1944 = 35/243. (-2x)^4 = 16x^4. Product: 35/243 * 16 = 560/243. Positive. The stated answer is correct. The solution steps are correct. Is there a style issue? The sentence is a bit dense but correct. The limit check is a valid verification. I will mark it as pass.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the binomial series formula and the appropriate substitutions for r and u. The verification step confirms the coefficients match the function expansion.gpt-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 2026-10-17 with SymPy 1.14.0.