Taylor series about a point
Problem 7.314 · medium
Find the Taylor polynomial of degree 3 for \( \displaystyle f(x) = \cos{\left(x \right)} \) centered at \( \displaystyle a = \pi \).
- \[ \left[\begin{matrix}\left. \cos{\left(x \right)} \right|_{\substack{ x=\pi }}\\\left. \frac{d}{d x} \cos{\left(x \right)} \right|_{\substack{ x=\pi }}\\\left. \frac{d^{2}}{d x^{2}} \cos{\left(x \right)} \right|_{\substack{ x=\pi }}\\\left. \frac{d^{3}}{d x^{3}} \cos{\left(x \right)} \right|_{\substack{ x=\pi }}\end{matrix}\right] = \left[\begin{matrix}-1\\0\\1\\0\end{matrix}\right] \]f and its first three derivatives at a.✓ Proved
- T(x) = Σ f⁽ʲ⁾(a)/j! · (x − a)ʲ.
- \[ \lim_{x \to \pi^+}\left(\frac{- \frac{\left(x - \pi\right)^{2}}{2} + \cos{\left(x \right)} + 1}{\left(x - \pi\right)^{3}}\right) = 0 \]T agrees with f to third order at a.✓ Proved
Answer \( T_3(x) = \frac{\left(x - \pi\right)^{2}}{2} - 1 \)
Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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: fail (style) — The solution verifies the remainder but never writes the explicit degree‑3 Taylor polynomial; the conclusion is missing, so the answer is incomplete.qwen3.6:27b-mlx: pass — The solution correctly identifies the derivatives and constructs the Taylor polynomial. The limit check confirms the third-order agreement, validating the result.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the derivatives and constructs the Taylor polynomial. The limit check confirms the third-order agreement, validating the result.gpt-oss:20b: fail (style) 2026-10-05 — The solution verifies the remainder but never writes the explicit degree‑3 Taylor polynomial; the conclusion is missing, so the answer is incomplete.qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the derivatives and constructs the Taylor polynomial. The limit check confirms the remainder is of order 3, validating the degree 3 approximation.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.