The binomial series
Problem 7.661 · medium
Find the first five terms of the binomial series for \( \displaystyle \sqrt{1 - 2 x} \).
- (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.
- \[ \frac{\left(-2\right)^{2} \left(-1 + 1 \cdot \frac{1}{2}\right)}{2 \cdot 2} = - \frac{1}{2} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{\frac{5 x^{4}}{8} + \frac{x^{3}}{2} + \frac{x^{2}}{2} + x + \sqrt{1 - 2 x} - 1}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( - \frac{5 x^{4}}{8} - \frac{x^{3}}{2} - \frac{x^{2}}{2} - x + 1 + \cdots \)
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: fail (misleading) — The solution lists the binomial expansion with r=½ and u=−2x correctly, but it only verifies the x² coefficient and then uses a limit with the wrong signs (+ x, + x²/2, + x³/2, + 5x⁴/8). This misleads a student into thinking the series has positive terms, whereas the correct first five terms are 1 − x − x²/2 − x³/2 − 5x⁴/8. The conclusion that the terms agree to order x⁴ is therefore false.qwen3.6:27b-mlx: pass — The solution correctly identifies the binomial series formula and parameters. The algebraic check for the x^2 coefficient is correct, and the limit check confirms the expansion matches the function up to the required order.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the binomial series formula and parameters. The algebraic check for the x^2 coefficient is correct, and the limit check confirms the expansion matches the function up to the required order.gpt-oss:20b: fail (misleading) 2026-10-18 — The solution lists the binomial expansion with r=½ and u=−2x correctly, but it only verifies the x² coefficient and then uses a limit with the wrong signs (+ x, + x²/2, + x³/2, + 5x⁴/8). This misleads a student into thinking the series has positive terms, whereas the correct first five terms are 1 − x − x²/2 − x³/2 − 5x⁴/8. The conclusion that the terms agree to order x⁴ is therefore false.qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the binomial series formula and parameters. The algebraic verification of the coefficient and the limit check confirm the result matches the stated answer.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the binomial expansion formula and the substitution values. The algebraic checks confirm the coefficients match the stated answer.gpt-oss:20b: pass 2026-10-15
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 2026-10-18 with SymPy 1.14.0.