The binomial series
Problem 7.715 · medium
Find the first five terms of the binomial series for \( \displaystyle \sqrt{3 x + 1} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = 1/2 and u = 3*x.
- \[ \frac{3^{2} \left(-1 + 1 \cdot \frac{1}{2}\right)}{2 \cdot 2} = - \frac{9}{8} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{\frac{405 x^{4}}{128} - \frac{27 x^{3}}{16} + \frac{9 x^{2}}{8} - \frac{3 x}{2} + \sqrt{3 x + 1} - 1}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( - \frac{405 x^{4}}{128} + \frac{27 x^{3}}{16} - \frac{9 x^{2}}{8} + \frac{3 x}{2} + 1 + \cdots \)
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 (style) — The solution correctly sets up the binomial expansion and verifies the x² coefficient, but it does not actually list the first five terms of the series as requested. The missing terms make the answer incomplete rather than incorrect.qwen3.6:27b-mlx: fail (error) — The stated answer lists terms in descending order of power, but the coefficient for x^4 is negative (-405/128), whereas the binomial expansion for (1+3x)^(1/2) yields a positive coefficient (+405/128). The solution's verification in line 3 checks against the wrong sign (subtracting 405/128), confirming the error in the stated answer rather than correcting it.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The stated answer lists terms in descending order of power, but the coefficient for x^4 is negative (-405/128), whereas the binomial expansion for (1+3x)^(1/2) yields a positive coefficient (+405/128). The solution's verification in line 3 checks against the wrong sign (subtracting 405/128), confirming the error in the stated answer rather than correcting it.gpt-oss:20b: fail (style) 2026-10-19 — The solution correctly sets up the binomial expansion and verifies the x² coefficient, but it does not actually list the first five terms of the series as requested. The missing terms make the answer incomplete rather than incorrect.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The stated answer lists the terms in descending order of powers, which is non-standard for a series expansion. More critically, the coefficients in the stated answer have incorrect signs for the $x^3$ and $x^4$ terms (the binomial expansion of $(1+3x)^{1/2}$ alternates signs starting with positive for $x$, so $x^2$ is negative, $x^3$ is positive, $x^4$ is negative; the answer has $x^3$ positive and $x^4$ negative, but the magnitude of the $x^4$ term in the answer is $-405/128$, whereas the correct coefficient is $-405/128$? Let's check: $c_4 = \binom{1/2}{4} 3^4 = \frac{(1/2)(-1/2)(-3/2)(-5/2)}{24} 81 = \frac{-15/16}{24} 81 = \frac{-5}{128} 81 = -405/128$. The sign is correct. Wait, let's re-read the stated answer: $- \frac{405 x^{4}}{128} + \frac{27 x^{3}}{16} - \frac{9 x^{2}}{8} + \frac{3 x}{2} + 1$. The signs are: $x^4$ (-), $x^3$ (+), $x^2$ (-), $x$ (+), const (+). This matches the alternating pattern. However, the solution text only verifies the $x^2$ coefficient and the limit for the whole polynomial. It does not explicitly derive or verify the $x^3$ and $x^4$ coefficients in the text, relying on the 'proved' equation which checks the limit. The limit check confirms the polynomial is correct up to $O(x^4)$. So the math is correct. Why fail? The prompt asks to find the first five terms. The solution provides them. The setup is correct. The verification is correct. Is there a subtle error? The stated answer writes the series in descending order. This is unusual but not wrong. Let's look closer at the limit equation. `Limit((405*x**4/128 - 27*x**3/16 + 9*x**2/8 - 3*x/2 + sqrt(3*x + 1) - 1)/x**4, x, 0, dir='+-') = 0`. This checks that `sqrt(3x+1) - 1 - (3x/2 - 9x^2/8 + 27x^3/16 - 405x^4/128)` is $o(x^4)$. This confirms the coefficients are correct. The solution is sound. Wait, I need to be adversarial. Is the order of terms in the stated answer considered 'wrong' in a strict pedagogical sense? Usually, series are written in ascending order. Writing them in descending order is confusing but not mathematically false. However, look at the first sentence: `u = 3*x`. The expansion is for `(1+3x)^(1/2)`. The problem is `sqrt(3x+1)`. This is correct. The coefficients are correct. The limit check is correct. I will mark it as pass.gpt-oss:20b: inconclusive 2026-10-16 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The limit test in sentence 3 uses the wrong signs for the series terms; the coefficient of \(x^4\) should be \(-405/128\), not \(+405/128\)."}qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer contains incorrect signs for the even-powered terms (x^2 and x^4). The binomial expansion of (1+u)^(1/2) has alternating signs, sogpt-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.