Taylor series about a point
Problem 7.316 · medium
Find the Taylor polynomial of degree 3 for \( \displaystyle f(x) = x^{3} - 2 x \) centered at \( \displaystyle a = 1 \).
- \[ \left[\begin{matrix}\left. x^{3} - 2 x \right|_{\substack{ x=1 }}\\\left. \frac{d}{d x} \left(x^{3} - 2 x\right) \right|_{\substack{ x=1 }}\\\left. \frac{d^{2}}{d x^{2}} \left(x^{3} - 2 x\right) \right|_{\substack{ x=1 }}\\\left. \frac{d^{3}}{d x^{3}} \left(x^{3} - 2 x\right) \right|_{\substack{ x=1 }}\end{matrix}\right] = \left[\begin{matrix}-1\\1\\6\\6\end{matrix}\right] \]f and its first three derivatives at a.✓ Proved
- T(x) = Σ f⁽ʲ⁾(a)/j! · (x − a)ʲ.
- \[ \lim_{x \to 1^+}\left(\frac{x^{3} - 3 x - \left(x - 1\right)^{3} - 3 \left(x - 1\right)^{2} + 2}{\left(x - 1\right)^{3}}\right) = 0 \]T agrees with f to third order at a.✓ Proved
Answer \( T_3(x) = x + \left(x - 1\right)^{3} + 3 \left(x - 1\right)^{2} - 2 \)
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.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | a sentence; read, not computed |
| 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 | SymPy's series() expansion about a |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The stated answer is incorrect; the constant term should be -1, not -2. Additionally, the verification in line 3 checks a polynomial with a constant term of +2, which does not match the stated answer (constant term -2) nor the correct Taylor polynomial.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The stated answer is incorrect; the constant term should be -1, not -2. Additionally, the verification in line 3 checks a polynomial with a constant term of +2, which does not match the stated answer (constant term -2) nor the correct Taylor polynomial.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The stated answer is incorrect. The correct Taylor polynomial is T_3(x) = -1 + (x-1) + 3(x-1)^2 + (x-1)^3, but the provided answer has a constant term of -2 and a linear term of x (which implies a constant of -1 plus a linear shift, but the coefficients are mixed up). Specifically, f(1) = -1, so the constant term must be -1, not -2. The solution's verification step (line 3) checks a different polynomial than the stated answer, failing to validate the final result.gpt-oss:20b: pass 2026-10-05
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/taylor_series_center, checked 2026-10-05 with SymPy 1.14.0.