Taylor series about a point
Problem 7.313 · hard
Find the Taylor polynomial of degree 3 for \( \displaystyle f(x) = \ln{\left(x \right)} \) centered at \( \displaystyle a = 2 \).
- \[ \left[\begin{matrix}\left. \ln{\left(x \right)} \right|_{\substack{ x=2 }}\\\left. \frac{d}{d x} \ln{\left(x \right)} \right|_{\substack{ x=2 }}\\\left. \frac{d^{2}}{d x^{2}} \ln{\left(x \right)} \right|_{\substack{ x=2 }}\\\left. \frac{d^{3}}{d x^{3}} \ln{\left(x \right)} \right|_{\substack{ x=2 }}\end{matrix}\right] = \left[\begin{matrix}\ln{\left(2 \right)}\\\frac{1}{2}\\- \frac{1}{4}\\\frac{1}{4}\end{matrix}\right] \]f and its first three derivatives at a.✓ Proved
- T(x) = Σ f⁽ʲ⁾(a)/j! · (x − a)ʲ.
- \[ \lim_{x \to 2^+}\left(\frac{- \frac{x}{2} - \frac{\left(x - 2\right)^{3}}{24} + \frac{\left(x - 2\right)^{2}}{8} + \ln{\left(x \right)} - \ln{\left(2 \right)} + 1}{\left(x - 2\right)^{3}}\right) = 0 \]T agrees with f to third order at a.✓ Proved
Answer \( T_3(x) = \frac{x}{2} + \frac{\left(x - 2\right)^{3}}{24} - \frac{\left(x - 2\right)^{2}}{8} - 1 + \ln{\left(2 \right)} \)
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 linear term should be (x-2)/2, not x/2. The solution fails to explicitly construct the polynomial from the derivatives, instead relying on a limit check that does not verify the specific coefficients of the stated answer.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The stated answer is incorrect; the linear term should be (x-2)/2, not x/2. The solution fails to explicitly construct the polynomial from the derivatives, instead relying on a limit check that does not verify the specific coefficients of the stated answer.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: inconclusive 2026-10-05 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution computes the derivatives correctly but fails to divide by the factorial terms (j!) when constructing the polynomial coefficients. For exgpt-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.