The binomial series
Problem 7.712 · medium
Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt{4 x + 1}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -1/2 and u = 4*x.
- \[ \frac{4^{2} \left(1 \cdot \frac{1}{2} + 1\right)}{2 \cdot 2} = 6 \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- 70 x^{4} + 20 x^{3} - 6 x^{2} + 2 x - 1 + \frac{1}{\sqrt{4 x + 1}}}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( 70 x^{4} - 20 x^{3} + 6 x^{2} - 2 x + 1 + \cdots \)
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: fail (error) — The stated answer has incorrect signs for all non-constant terms. The binomial expansion of (1+4x)^(-1/2) yields 1 - 2x + 6x^2 - 20x^3 + 70x^4, whereas the stated answer lists 1 - 2x + 6x^2 - 20x^3 + 70x^4 but with alternating signs flipped for the higher order terms in the provided text (specifically, the stated answer shows -20x^3 and +70x^4, but the limit check in line 3 uses -70x^4 + 20x^3... wait, let's look closer. The stated answer is 70x^4 - 20x^3 + 6x^2 - 2x + 1. The correct expansion is 1 - 2x + 6x^2 - 20x^3 + 70x^4. The stated answer matches the correct coefficients and signs. However, line 3 checks `(-70*x**4 + 20*x**3 - 6*x**2 + 2*x - 1 + 1/sqrt(4*x + 1))/x**4`. This expression is `-(70x^4 - 20x^3 + 6x^2 - 2x + 1) + f(x)`. If the limit is 0, then `f(x) ~ 70x^4 - 20x^3 + 6x^2 - 2x + 1`. This matches the stated answer. So the solution is actually correct? Let me re-read the stated answer carefully. `70 x^{4} - 20 x^{3} + 6 x^{2} - 2 x + 1`. Correct terms: `1`, `-2x`, `+6x^2`, `-20x^3`, `+70x^4`. The stated answer has `+70x^4`, `-20x^3`, `+6x^2`, `-2x`, `+1`. This is correct. Why did I think it was wrong? Ah, I misread the signs in my head. The solution is correct. Wait, let me double check the coefficient calculation. r=-1/2. Term 1: 1. Term 2: (-1/2)(4x) = -2x. Term 3: (-1/2)(-3/2)/2 * (4x)^2 = (3/8)*16x^2 = 6x^2. Term 4: (-1/2)(-3/2)(-5/2)/6 * (4x)^3 = (-15/8)/6 * 64x^3 = (-15/48)*64x^3 = -20x^3. Term 5: (-1/2)(-3/2)(-5/2)(-7/2)/24 * (4x)^4 = (105/16)/24 * 256x^4 = (105/384)*256x^4 = 70x^4. The stated answer is correct. The limit check in line 3 confirms the polynomial `70x^4 - 20x^3 + 6x^2 - 2x + 1` approximates the function. The solution is sound.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The stated answer has incorrect signs for all non-constant terms. The binomial expansion of (1+4x)^(-1/2) yields 1 - 2x + 6x^2 - 20x^3 + 70x^4, whereas the stated answer lists 1 - 2x + 6x^2 - 20x^3 + 70x^4 but with alternating signs flipped for the higher order terms in the provided text (specifically, the stated answer shows -20x^3 and +70x^4, but the limit check in line 3 uses -70x^4 + 20x^3... wait, let's look closer. The stated answer is 70x^4 - 20x^3 + 6x^2 - 2x + 1. The correct expansion is 1 - 2x + 6x^2 - 20x^3 + 70x^4. The stated answer matches the correct coefficients and signs. However, line 3 checks `(-70*x**4 + 20*x**3 - 6*x**2 + 2*x - 1 + 1/sqrt(4*x + 1))/x**4`. This expression is `-(70x^4 - 20x^3 + 6x^2 - 2x + 1) + f(x)`. If the limit is 0, then `f(x) ~ 70x^4 - 20x^3 + 6x^2 - 2x + 1`. This matches the stated answer. So the solution is actually correct? Let me re-read the stated answer carefully. `70 x^{4} - 20 x^{3} + 6 x^{2} - 2 x + 1`. Correct terms: `1`, `-2x`, `+6x^2`, `-20x^3`, `+70x^4`. The stated answer has `+70x^4`, `-20x^3`, `+6x^2`, `-2x`, `+1`. This is correct. Why did I think it was wrong? Ah, I misread the signs in my head. The solution is correct. Wait, let me double check the coefficient calculation. r=-1/2. Term 1: 1. Term 2: (-1/2)(4x) = -2x. Term 3: (-1/2)(-3/2)/2 * (4x)^2 = (3/8)*16x^2 = 6x^2. Term 4: (-1/2)(-3/2)(-5/2)/6 * (4x)^3 = (-15/8)/6 * 64x^3 = (-15/48)*64x^3 = -20x^3. Term 5: (-1/2)(-3/2)(-5/2)(-7/2)/24 * (4x)^4 = (105/16)/24 * 256x^4 = (105/384)*256x^4 = 70x^4. The stated answer is correct. The limit check in line 3 confirms the polynomial `70x^4 - 20x^3 + 6x^2 - 2x + 1` approximates the function. The solution is sound.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (style) 2026-10-16 — [domain objection, downgraded to style] The stated answer has incorrect signs for the terms (the binomial series for (1+4x)^{-1/2} has alternating signs starting with +1, -2x, +6x^2, etc., but the provided answer lists them as 1, -2x, +6x^2, -20x^3, +70x^4, which is actually correct in sign pattern, wait. Let's re-evaluate. Binomial expansion of (1+u)^{-1/2}: 1 + (-1/2)u + (-1/2)(-3/2)/2! u^2 + (-1/2)(-3/2)(-5/2)/3! u^3 + (-1/2)(-3/2)(-5/2)(-7/2)/4! u^4 u = 4x Term 0: 1 Term 1: -1/2 * 4x = -2x Term 2: (3/8) * (4x)^2 = (3/8)*16x^2 = 6x^2 Term 3: (-15/48) * (4x)^3 = (-5/16) * 64x^3 = -20x^3 Term 4: (105/384) * (4x)^4 = (35/128) * 256x^4 = 70x^4 So the series is 1 - 2x + 6x^2 - 20x^3 + 70x^4. The stated answer is: 70 x^{4} - 20 x^{3} + 6 x^{2} - 2 x + 1. This is the same polynomial, just written in descending order. So the stated answer is correct. Let's look at the solution steps. Step 1: Sets up the formula correctly. Step 2: Checks the x^2 coefficient. It says `(-(-1 - 1/2)/2/2)*4**2 = 6`. Let's parse the coefficient calculation: r(r-1)/2! * u^2. r = -1/2. r-1 = -3/2. r(r-1)/2 = (-1/2)(-3/2)/2 = (3/4)/2 = 3/8. 3/8 * (4x)^2 = 3/8 * 16 x^2 = 6 x^2. The code snippet `(-(-1 - 1/2)/2/2)` evaluates to: -1 - 1/2 = -1.5 -(-1.5) = 1.5 1.5 / 2 = 0.75 0.75 / 2 = 0.375 = 3/8. So the coefficient calculation is correct. Step 3: Checks the limit of the difference divided by x^4. The expression is `(-70*x**4 + 20*x**3 - 6*x**2 + 2*x - 1 + 1/sqrt(4*x + 1))/x**4`. Note the signs inside the numerator: `-70*x**4 + 20*x**3 - 6*x**2 + 2*x - 1`. This corresponds to subtracting the polynomial `70x^4 - 20x^3 + 6x^2 - 2x + 1` from the function. If the polynomial is P(x) = 1 - 2x + 6x^2 - 20x^3 + 70x^4, then P(x) = 70x^4 - 20x^3 + 6x^2 - 2x + 1. The limit checks if f(x) - P(x) is o(x^4). The limit is 0, which confirms the expansion is correct up to x^4. The solution seems correct. Why would it fail? Let's re-read the prompt constraints. "Find the first five terms". The terms are 1, -2x, 6x^2, -20x^3, 70x^4. The stated answer lists them. The solution steps verify the setup and the coefficients. Is there a subtle error? Step 1 says `u = 4*x`. The function is `1/sqrt(4x+1) = (1+4x)^{-1/2}`. This is correct. Step 2 calculates the x^2 coefficient. It is correct. Step 3 verifies the whole polynomial. It is correct. Wait, look at Step 2's comment: `-- The x² coefficient, r(r − 1)/2 · c².` The variable used in the comment is `c`. In the equation, it uses `4`. `c` is not defined in the sentence, but `4` is the coefficient of x in `u=4x`. This is a minor style issue, not an error. Is the verdict pass? Let's check the signs in Step 3 again. `-70*x**4 + 20*x**3 - 6*x**2 + 2*x - 1` This is `-(70x^4 - 20x^3 + 6x^2 - 2x + 1)`. So the numerator is `f(x) - P(x)`. Limit is 0. Correct. I see no error. The solution is sound.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (style) 2026-10-16 — [domain objection, downgraded to style] The stated answer has incorrect signs for the terms (all terms in the expansion of (1+4x)^(-1/2) alternate in sign, starting with +1, so the x term should be -2x, x^2 term +6x^2, etc., but the provided answer lists them as -2x, +6x^2, -20x^3, +70x^4 which is actually correct in sign pattern but the solution text claims the limit check confirms the terms. Wait, let's look closer. The expansion is 1 - 2x + 6x^2 - 20x^3 + 70x^4. The stated answer is 70x^4 - 20x^3 + 6x^2 - 2x + 1. This is correct. However, the solution only calculates the x^2 coefficient explicitly and then asserts the rest via a limit check without showing the derivation for the other terms. More critically, the prompt asks for a review of the *solution*. The solution is incomplete because it doesn't derive the other coefficients, but the main issue is usually logical flow. Let's look at the limit check. The limit checks if the polynomial matches the function. It does. Is there an error? The problem asks for the first five terms. The solution provides them. The derivation is sparse. But is it wrong? No. Is it misleading? It skips steps. But usually 'error' is for wrong math. Let's re-read the prompt constraints. 'Find what is wrong'. The solution is technically correct in its assertions. However, standard binomial series solutions should show the calculation of coefficients. The provided solution is a 'black box' verification. Is that an error? No. Is it misleading? Maybe. Let's look for a subtle error. r = -1/2. u = 4x. Term 1: 1. Term 2: (-1/2)(4x) = -2x. Term 3: (-1/2)(-3/2)/2 * (4x)^2 = (3/8)*16x^2 = 6x^2. Term 4: (-1/2)(-3/2)(-5/2)/6 * (4x)^3 = (-15/8)/6 * 64x^3 = (-5/16)*64x^3 = -20x^3. Term 5: (-1/2)(-3/2)(-5/2)(-7/2)/24 * (4x)^4 = (105/16)/24 * 256x^4 = (105/384)*256x^4 = 70x^4. The coefficients are correct. The signs are correct. The limit check is valid. Why would it fail? Perhaps the 'stated answer' order? Usually series are written in ascending powers. The stated answer is descending. This is non-standard but not wrong. However, looking at the provided solution text again: 'Limit(..., x, 0, dir='+-') = 0'. This confirms the polynomial is the Taylor polynomial. The solution is sound. Wait, I must find what is wrong. If nothing is wrong, I say pass. But the prompt implies there might be an error. Let's check the limit expression carefully. `(-70*x**4 + 20*x**3 - 6*x**2 + 2*x - 1 + 1/sqrt(4*x + 1))/x**4`. The polynomial part is `-70x^4 + 20x^3 - 6x^2 + 2x - 1`. This is the negative of the series `1 - 2x + 6x^2 - 20x^3 + 70x^4`. So the numerator is `-(Series) + Function`. If Series approximates Function, this goes to 0. This is correct. Is there a domain issue? The binomial series converges for |4x| < 1, i.e., |x| < 1/4. The limit as x->0 is within the radius of convergence. Everything seems correct. I will mark it as pass.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.