Taylor series about a point
Problem 7.315 · medium
Find the Taylor polynomial of degree 3 for \( \displaystyle f(x) = \frac{1}{x} \) centered at \( \displaystyle a = 1 \).
- \[ \left[\begin{matrix}\left. \frac{1}{x} \right|_{\substack{ x=1 }}\\\left. \frac{d}{d x} \frac{1}{x} \right|_{\substack{ x=1 }}\\\left. \frac{d^{2}}{d x^{2}} \frac{1}{x} \right|_{\substack{ x=1 }}\\\left. \frac{d^{3}}{d x^{3}} \frac{1}{x} \right|_{\substack{ x=1 }}\end{matrix}\right] = \left[\begin{matrix}1\\-1\\2\\-6\end{matrix}\right] \]f and its first three derivatives at a.✓ Proved
- T(x) = Σ f⁽ʲ⁾(a)/j! · (x − a)ʲ.
- \[ \lim_{x \to 1^+}\left(\frac{x + \left(x - 1\right)^{3} - \left(x - 1\right)^{2} - 2 + \frac{1}{x}}{\left(x - 1\right)^{3}}\right) = 0 \]T agrees with f to third order at a.✓ Proved
Answer \( T_3(x) = - x - \left(x - 1\right)^{3} + \left(x - 1\right)^{2} + 2 \)
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 T_3(x) = -x - (x-1)^3 + (x-1)^2 + 2 is incorrect. The correct Taylor polynomial is 1 - (x-1) + (x-1)^2 - (x-1)^3. The provided solution fails to explicitly construct the polynomial from the derivatives in step 1, and the 'proof' in step 3 checks a different function (which simplifies to 1/x - 1 + (x-1)^2 - (x-1)^3, missing the linear term and having wrong signs) rather than the stated answer.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The stated answer T_3(x) = -x - (x-1)^3 + (x-1)^2 + 2 is incorrect. The correct Taylor polynomial is 1 - (x-1) + (x-1)^2 - (x-1)^3. The provided solution fails to explicitly construct the polynomial from the derivatives in step 1, and the 'proof' in step 3 checks a different function (which simplifies to 1/x - 1 + (x-1)^2 - (x-1)^3, missing the linear term and having wrong signs) rather than the stated answer.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The stated answer T_3(x) = -x - (x - 1)^3 + (x - 1)^2 + 2 is incorrect. The correct Taylor polynomial is 1 - (x - 1) + (x - 1)^2 - (x - 1)^3. The provided solution fails to construct the polynomial from the derivatives calculated in step 1, and step 3 verifies an incorrect expression.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.