The binomial series
Problem 7.710 · medium
Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt{x + 1}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -1/2 and u = x.
- \[ \frac{1^{2} \left(1 \cdot \frac{1}{2} + 1\right)}{2 \cdot 2} = \frac{3}{8} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- \frac{35 x^{4}}{128} + \frac{5 x^{3}}{16} - \frac{3 x^{2}}{8} + \frac{x}{2} - 1 + \frac{1}{\sqrt{x + 1}}}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( \frac{35 x^{4}}{128} - \frac{5 x^{3}}{16} + \frac{3 x^{2}}{8} - \frac{x}{2} + 1 + \cdots \)
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: fail (error) — The solution only verifies the x² coefficient and a limit test; it never actually lists the first five terms of the series, nor does it compute the x³ and x⁴ coefficients. The setup is correct, but the conclusion is incomplete and misleading.qwen3.6:27b-mlx: fail (error) — The stated answer has incorrect signs for all variable terms (it lists alternating signs starting with positive for x^2, but the binomial expansion for (1+x)^{-1/2} has all negative coefficients for x^n where n >= 1). The solution setup correctly identifies r=-1/2, but the final result provided contradicts the standard expansion 1 - x/2 + 3x^2/8 - 5x^3/16 + 35x^4/128. Specifically, the stated answer has +3x^2/8 (correct) but -x/2 (correct) and -5x^3/16 (correct) and +35x^4/128 (correct)... wait, let me re-read the stated answer carefully. Stated answer: 35x^4/128 - 5x^3/16 + 3x^2/8 - x/2 + 1. Standard expansion: 1 - 1/2 x + 3/8 x^2 - 5/16 x^3 + 35/128 x^4. Comparing term by term: Constant: 1 vs 1 (Match) x term: -x/2 vs -x/2 (Match) x^2 term: 3x^2/8 vs 3x^2/8 (Match) x^3 term: -5x^3/16 vs -5x^3/16 (Match) x^4 term: 35x^4/128 vs 35x^4/128 (Match) The stated answer is actually correct. The terms are just written in descending order of power. Let's re-evaluate the solution steps. Step 1: Sets up the binomial series formula correctly with r=-1/2 and u=x. Step 2: Checks the x^2 coefficient. The calculation shown is `(-(-1 - 1/2)/2/2)*1**2 = 3/8`. Let's trace this. r = -1/2. r(r-1)/2 = (-1/2)(-3/2)/2 = (3/4)/2 = 3/8. The code snippet `(-(-1 - 1/2)/2/2)` seems to be a garbled attempt to calculate `r(r-1)/2`. `-1 - 1/2` is `-3/2`. Negated is `3/2`. Divided by 2 is `3/4`. Divided by 2 is `3/8`. So the arithmetic is correct for the coefficient. Step 3: Checks the limit of the difference between the function and the polynomial divided by x^4. The polynomial in the limit is `-35*x**4/128 + 5*x**3/16 - 3*x**2/8 + x/2 - 1`. This is the negative of the stated answer's variable part plus 1? No, the stated answer is `35 x^{4}/128 - 5 x^{3}/16 + 3 x^{2}/8 - x/2 + 1`. The limit checks `1/sqrt(x+1) - (1 - x/2 + 3x^2/8 - 5x^3/16 + 35x^4/128)`. The expression in the limit is `(-35*x**4/128 + 5*x**3/16 - 3*x**2/8 + x/2 - 1) + 1/sqrt(x + 1)`. This is equivalent to `1/sqrt(x+1) - (1 - x/2 + 3x^2/8 - 5x^3/16 + 35x^4/128)`. This matches the standard expansion. The limit being 0 confirms the coefficients are correct. The solution is sound. The stated answer is correct. The steps verify the coefficients. Wait, look at Step 2 again. `(-(-1 - 1/2)/2/2)*1**2 = 3/8`. The comment says `r(r − 1)/2 · c²`. Here c is likely 1 (coefficient of u). The calculation is correct. Is there any error? The problem asks for the first five terms. The stated answer has 5 terms. The solution verifies the x^2 term and the overall fit up to x^4. It seems correct. Why did I think it was wrong initially? I misread the signs in the stated answer as alternating starting with positive for the highest power, but they are consistent with the standard expansion. Let's double check the signs in the stated answer: `35 x^{4}/128` (positive) `- 5 x^{3}/16` (negative) `+ 3 x^{2}/8` (positive) `- x/2` (negative) `+ 1` (positive) Standard expansion for (1+x)^{-1/2}: 1 - 1/2 x + (-1/2)(-3/2)/2 x^2 = 3/8 x^2 + (-1/2)(-3/2)(-5/2)/6 x^3 = -15/48 x^3 = -5/16 x^3 + (-1/2)(-3/2)(-5/2)(-7/2)/24 x^4 = 105/192 x^4 = 35/64 x^4? Wait. Binomial coeff for n=4: (-1/2)(-3/2)(-5/2)(-7/2) / 24 Numerator: 105 / 16 Denominator: 24 105 / (16*24) = 105 / 384. Divide by 3: 35 / 128. So the coefficient is +35/128. The stated answer is correct. The solution steps are correct. Verdict: pass.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The stated answer has incorrect signs for all variable terms (it lists alternating signs starting with positive for x^2, but the binomial expansion for (1+x)^{-1/2} has all negative coefficients for x^n where n >= 1). The solution setup correctly identifies r=-1/2, but the final result provided contradicts the standard expansion 1 - x/2 + 3x^2/8 - 5x^3/16 + 35x^4/128. Specifically, the stated answer has +3x^2/8 (correct) but -x/2 (correct) and -5x^3/16 (correct) and +35x^4/128 (correct)... wait, let me re-read the stated answer carefully. Stated answer: 35x^4/128 - 5x^3/16 + 3x^2/8 - x/2 + 1. Standard expansion: 1 - 1/2 x + 3/8 x^2 - 5/16 x^3 + 35/128 x^4. Comparing term by term: Constant: 1 vs 1 (Match) x term: -x/2 vs -x/2 (Match) x^2 term: 3x^2/8 vs 3x^2/8 (Match) x^3 term: -5x^3/16 vs -5x^3/16 (Match) x^4 term: 35x^4/128 vs 35x^4/128 (Match) The stated answer is actually correct. The terms are just written in descending order of power. Let's re-evaluate the solution steps. Step 1: Sets up the binomial series formula correctly with r=-1/2 and u=x. Step 2: Checks the x^2 coefficient. The calculation shown is `(-(-1 - 1/2)/2/2)*1**2 = 3/8`. Let's trace this. r = -1/2. r(r-1)/2 = (-1/2)(-3/2)/2 = (3/4)/2 = 3/8. The code snippet `(-(-1 - 1/2)/2/2)` seems to be a garbled attempt to calculate `r(r-1)/2`. `-1 - 1/2` is `-3/2`. Negated is `3/2`. Divided by 2 is `3/4`. Divided by 2 is `3/8`. So the arithmetic is correct for the coefficient. Step 3: Checks the limit of the difference between the function and the polynomial divided by x^4. The polynomial in the limit is `-35*x**4/128 + 5*x**3/16 - 3*x**2/8 + x/2 - 1`. This is the negative of the stated answer's variable part plus 1? No, the stated answer is `35 x^{4}/128 - 5 x^{3}/16 + 3 x^{2}/8 - x/2 + 1`. The limit checks `1/sqrt(x+1) - (1 - x/2 + 3x^2/8 - 5x^3/16 + 35x^4/128)`. The expression in the limit is `(-35*x**4/128 + 5*x**3/16 - 3*x**2/8 + x/2 - 1) + 1/sqrt(x + 1)`. This is equivalent to `1/sqrt(x+1) - (1 - x/2 + 3x^2/8 - 5x^3/16 + 35x^4/128)`. This matches the standard expansion. The limit being 0 confirms the coefficients are correct. The solution is sound. The stated answer is correct. The steps verify the coefficients. Wait, look at Step 2 again. `(-(-1 - 1/2)/2/2)*1**2 = 3/8`. The comment says `r(r − 1)/2 · c²`. Here c is likely 1 (coefficient of u). The calculation is correct. Is there any error? The problem asks for the first five terms. The stated answer has 5 terms. The solution verifies the x^2 term and the overall fit up to x^4. It seems correct. Why did I think it was wrong initially? I misread the signs in the stated answer as alternating starting with positive for the highest power, but they are consistent with the standard expansion. Let's double check the signs in the stated answer: `35 x^{4}/128` (positive) `- 5 x^{3}/16` (negative) `+ 3 x^{2}/8` (positive) `- x/2` (negative) `+ 1` (positive) Standard expansion for (1+x)^{-1/2}: 1 - 1/2 x + (-1/2)(-3/2)/2 x^2 = 3/8 x^2 + (-1/2)(-3/2)(-5/2)/6 x^3 = -15/48 x^3 = -5/16 x^3 + (-1/2)(-3/2)(-5/2)(-7/2)/24 x^4 = 105/192 x^4 = 35/64 x^4? Wait. Binomial coeff for n=4: (-1/2)(-3/2)(-5/2)(-7/2) / 24 Numerator: 105 / 16 Denominator: 24 105 / (16*24) = 105 / 384. Divide by 3: 35 / 128. So the coefficient is +35/128. The stated answer is correct. The solution steps are correct. Verdict: pass.gpt-oss:20b: fail (error) 2026-10-19 — The solution only verifies the x² coefficient and a limit test; it never actually lists the first five terms of the series, nor does it compute the x³ and x⁴ coefficients. The setup is correct, but the conclusion is incomplete and misleading.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The stated answer has the wrong signs for the odd-power terms (it lists -x/2 and -5x^3/16, but the expansion of (1+x)^{-1/2} has +x/2 and +5x^3/16). The solution's verification step (Line 3) confirms the terms with the correct signs, contradicting the stated answer.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer has incorrect signs for all non-constant terms. The binomial expansion of (1+x)^(-1/2) is 1 - x/2 + 3x^2/8 - 5x^3/16 + 35x^4/128,gpt-oss:20b: pass 2026-10-16
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 2026-10-19 with SymPy 1.14.0.