The binomial series
Problem 7.733 · medium
Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt{2 x + 1}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -1/2 and u = 2*x.
- \[ \frac{2^{2} \left(1 \cdot \frac{1}{2} + 1\right)}{2 \cdot 2} = \frac{3}{2} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- \frac{35 x^{4}}{8} + \frac{5 x^{3}}{2} - \frac{3 x^{2}}{2} + x - 1 + \frac{1}{\sqrt{2 x + 1}}}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( \frac{35 x^{4}}{8} - \frac{5 x^{3}}{2} + \frac{3 x^{2}}{2} - x + 1 + \cdots \)
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: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer has incorrect signs for the terms; the binomial expansion of (1+2x)^{-1/2} yields alternating signs starting with +1, -x, +3/2 x^2
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-23 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer has incorrect signs for the terms; the binomial expansion of (1+2x)^{-1/2} yields alternating signs starting with +1, -x, +3/2 x^2gpt-oss:20b: pass 2026-10-23qwen3.6:27b-mlx: fail (error) 2026-10-22 — The stated answer has incorrect signs for the odd-power terms (it lists -x and -5/2 x^3, but the expansion of (1+2x)^{-1/2} yields -x and +5/2 x^3). Additionally, the solution text claims the coefficient of x^2 is 3/2, but the stated answer lists 3/2 x^2, which is correct, yet the verification in line 3 uses a polynomial with -3/2 x^2, contradicting the stated answer and the correct expansion.gpt-oss:20b: pass 2026-10-22qwen3.6:27b-mlx: fail (error) 2026-10-22 — The stated answer has incorrect signs for all terms except the constant term. The binomial expansion of (1+2x)^(-1/2) yields 1 - x + (3/2)x^2 - (5/2)x^3 + (35/8)x^4, whereas the stated answer lists 1 - x + (3/2)x^2 - (5/2)x^3 + (35/8)x^4 but written in descending order with alternating signs that do not match the standard expansion (specifically, the x term should be negative, x^2 positive, x^3 negative, x^4 positive; the stated answer has x positive in the list '... - x + 1' which implies -x, but then '+ 3x^2/2' is correct, '- 5x^3/2' is correct, '+ 35x^4/8' is correct. Wait, let's re-read carefully. Stated: 35x^4/8 - 5x^3/2 + 3x^2/2 - x + 1. This is actually correct. 1 - x + 3/2 x^2 - 5/2 x^3 + 35/8 x^4. The stated answer is just written in descending powers. Let me re-evaluate. Expansion: 1 -1/2 * 2x = -x (-1/2)(-3/2)/2 * (2x)^2 = (3/8) * 4x^2 = 3/2 x^2 (-1/2)(-3/2)(-5/2)/6 * (2x)^3 = (-15/8)/6 * 8x^3 = -15/48 * 8x^3 = -5/2 x^3 (-1/2)(-3/2)(-5/2)(-7/2)/24 * (2x)^4 = (105/16)/24 * 16x^4 = 105/384 * 16x^4 = 35/8 x^4. So the series is 1 - x + 3/2 x^2 - 5/2 x^3 + 35/8 x^4. The stated answer is: 35/8 x^4 - 5/2 x^3 + 3/2 x^2 - x + 1. This matches perfectly. Why did I think it was wrong? I misread the signs in my head. Let's look at the solution steps. Step 1: Sets up formula correctly. Step 2: Checks x^2 coefficient. Calculation: r(r-1)/2 * c^2 where c=2. r=-1/2. (-1/2)(-3/2)/2 * 4 = (3/8)*4 = 3/2. Correct. Step 3: Checks limit of difference / x^4 is 0. This confirms the first 5 terms (up to x^4) are correct. The solution is actually correct. Wait, look at the prompt again. "Find the first five terms". Terms are: 1, -x, 3/2 x^2, -5/2 x^3, 35/8 x^4. The stated answer lists them. The solution verifies the setup and the coefficients. Is there any error? Step 1 says "u = 2*x". The function is 1/sqrt(2x+1) = (1+2x)^(-1/2). So u=2x is correct. Step 2 calculates the x^2 coefficient. Step 3 verifies the whole polynomial. It seems correct. Let me double check the "adversarial" part. Is there a subtle issue? "first five terms" usually implies non-zero terms. The terms are non-zero. Maybe the issue is with the notation in Step 2? "(-(-1 - 1/2)/2/2)*2**2" r = -1/2. r-1 = -3/2. r(r-1) = 3/4. Divided by 2! = 2. So 3/8. Times u^2 = (2x)^2 = 4x^2. Coefficient is 3/8 * 4 = 3/2. The code snippet: `(-(-1 - 1/2)/2/2)*2**2` `-1 - 1/2` is `-1.5`. `-(-1.5)` is `1.5`. `1.5 / 2` is `0.75`. `0.75 / 2` is `0.375` (which is 3/8). `3/8 * 4` is `1.5` (which is 3/2). The calculation is correct. Step 3 limit check is robust. I will mark it as pass.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.