Taylor series about a point
Problem 7.309 · hard
Find the Taylor polynomial of degree 3 for \( \displaystyle f(x) = e^{x} \) centered at \( \displaystyle a = 1 \).
- \[ \left[\begin{matrix}\left. e^{x} \right|_{\substack{ x=1 }}\\\left. \frac{d}{d x} e^{x} \right|_{\substack{ x=1 }}\\\left. \frac{d^{2}}{d x^{2}} e^{x} \right|_{\substack{ x=1 }}\\\left. \frac{d^{3}}{d x^{3}} e^{x} \right|_{\substack{ x=1 }}\end{matrix}\right] = \left[\begin{matrix}e\\e\\e\\e\end{matrix}\right] \]f and its first three derivatives at a.✓ Proved
- T(x) = Σ f⁽ʲ⁾(a)/j! · (x − a)ʲ.Reviewed
- \[ \lim_{x \to 1^+}\left(\frac{- \frac{e \left(x - 1\right)^{3}}{6} - \frac{e \left(x - 1\right)^{2}}{2} - e \left(x - 1\right) + e^{x} - e}{\left(x - 1\right)^{3}}\right) = 0 \]T agrees with f to third order at a.✓ Proved
Answer \( T_3(x) = \frac{e \left(x - 1\right)^{3}}{6} + \frac{e \left(x - 1\right)^{2}}{2} + e \left(x - 1\right) + e \)
Lines: 2 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly identifies the derivatives, applies the Taylor polynomial formula, and verifies the result. The final expression matches the stated answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the derivatives, applies the Taylor polynomial formula, and verifies the result. The final expression matches the stated answer.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution uses a matrix to list derivatives but fails to explicitly state the formula for the Taylor polynomial coefficients (dividing by j!). It jumps from the list of derivatives to the final polynomial without showing the calculation of the coefficients, making the derivation incomplete and potentially confusing for a student.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.