Home›Calculus 2›The binomial series›Problem 7.714

The binomial series

Problem 7.714 · medium

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(1 - 2 x\right)^{2}} \).

(1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -2 and u = -2*x.
\[ - \frac{2 \left(-2\right)^{2} \left(-2 - 1\right)}{2} = 12 \]
The x² coefficient, r(r − 1)/2 · c².✓ Proved
\[ \lim_{x \to 0}\left(\frac{- 80 x^{4} - 32 x^{3} - 12 x^{2} - 4 x - 1 + \frac{1}{\left(1 - 2 x\right)^{2}}}{x^{4}}\right) = 0 \]
These terms agree with the function to order x⁴.✓ Proved

Answer \( 80 x^{4} + 32 x^{3} + 12 x^{2} + 4 x + 1 + \cdots \)

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: pass
qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The setup incorrectly identifies u as -2x instead of -2x (which is technically correct for the form (1+u)^r if we write (1-2x) as (1+(-2x)), but the

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-19 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The setup incorrectly identifies u as -2x instead of -2x (which is technically correct for the form (1+u)^r if we write (1-2x) as (1+(-2x)), but the
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The setup incorrectly identifies u as -2x; for (1 - 2x)^{-2}, the binomial form (1+u)^r requires u = -2x, but the standard expansion is usually derived with u = -2x directly or by substituting -2x into (1+x)^r. However, the critical error is in the sign of the terms. The binomial series for (1-2x)^{-2} has all positive coefficients. The solution sets u = -2x, which is correct for the form (1+u)^r. Let's check the terms: r=-2, u=-2x. Term 1: 1. Term 2: -2(-2x) = 4x. Term 3: (-2)(-3)/2 * (-2x)^2 = 3 * 4x^2 = 12x^2. Term 4: (-2)(-3)(-4)/6 * (-2x)^3 = -4 * (-8x^3) = 32x^3. Term 5: (-2)(-3)(-4)(-5)/24 * (-2x)^4 = 5 * 16x^4 = 80x^4. The coefficients 1, 4, 12, 32, 80 are correct. The stated answer lists them in descending order of power, which is unusual but not wrong. Wait, looking at the sentence: 'u = -2*x'. This is correct. The equation checks are correct. Why would it fail? Let's re-read carefully. 'Find the first five terms'. The answer provides 5 terms. The setup is correct. The checks are correct. Is there a subtle error? The prompt asks for the first five terms. The answer gives 80x^4 + ... + 1. This is the polynomial. The sentence says 'u = -2*x'. This is correct. The equation checks the coefficient of x^2 and the limit for x^4. Everything seems correct. Let me look for a 'misleading' aspect. The solution jumps from the general formula to checking specific coefficients without showing the derivation of the other terms, but the limit check confirms the whole polynomial. Is the limit check sufficient? It checks that the difference is o(x^4), which implies the coefficients up to x^4 are correct. This is a valid verification. However, the sentence 'u = -2*x' is technically part of the substitution. The problem is (1-2x)^{-2}. This is (1 + (-2x))^{-2}. So u = -2x. This is correct. I will mark it as pass.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The setup incorrectly identifies u as -2x. For the function (1 - 2x)^{-2}, the binomial expansion form (1 + u)^r requires u = -2x, but the standard substitution is usually written as (1 + (-2x))^r. While u = -2x is technically correct for the form (1+u)^r, the solution fails to explicitly state the resulting series terms or the general term derivation, jumping straight to coefficient checks. More critically, the stated answer lists terms in descending order of power (x^4 down to constant), which is non-standard and potentially confusing, but the main error is that the solution does not actually *derive* the first five terms as requested, only verifies the x^4 coefficient and the limit. It does not provide the terms 1, 4x, 12x^2, 32x^3, 80x^4 in a clear list derived from the formula. However, looking closer at the prompt's constraints: 'Find the first five terms'. The solution provides a formula and checks one coefficient and a limit. It does not explicitly list the first five terms as the final output of the derivation steps, although the limit check implies them. But wait, the 'Stated answer' is provided separately. The solution text itself is just the setup and checks. Is the setup wrong? u = -2x is correct for (1-2x). r = -2 is correct. The formula is correct. The checks are correct. Why fail? The solution does not actually *find* the terms in the text, it just verifies them. But usually, these reviews check if the logic is sound. Let's look at the 'Stated answer' vs the problem. The problem asks for the first five terms. The stated answer is `80 x^{4} + 32 x^{3} + 12 x^{2} + 4 x + 1`. This is correct. The solution steps: 1. Sets up formula. 2. Checks x^2 coeff. 3. Checks limit for x^4. It misses checking x^1 and x^3 explicitly, but the limit check covers all up to x^4. Is it misleading? It's incomplete as a 'solution' because it doesn't show the calculation for all terms, but the logic is sound. However, there is a subtle issue: The binomial series for (1-2x)^{-2} is sum (n+1)(2x)^n. Terms: 1, 4x, 12x^2, 32x^3, 80x^4. The solution says u = -2x. Then the term is r(r-1).../n! * u^n. For n=1: -2 * (-2x) = 4x. Correct. For n=2: (-2)(-3)/2 * (-2x)^2 = 3 * 4x^2 = 12x^2. Correct. The solution is technically correct but very sparse. Let's look for a real error. Ah, the stated answer in the prompt is `80 x^{4} + 32 x^{3} + 12 x^{2} + 4 x + 1`. The solution does not explicitly state this list. But the instructions say 'Review worked calculus solutions'. If the solution doesn't produce the answer, is it a fail? Usually, yes. But let's look at the 'Stated answer' field. That is the target. The solution is the work. The work is incomplete. It only checks two things. It doesn't derive the other three. A student reading this would not know how to get the x, x^3, or constant terms from the text provided. It's misleading/incomplete. Let's call it 'error' because it fails to solve the problem as stated (finding all five terms).
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.