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Maclaurin series by substitution

Problem 7.273 · medium

Use a known Maclaurin series to find the first four nonzero terms of the Maclaurin series of \( \displaystyle f(x) = e^{- x^{2}} \).

Start from a known series (e^u with u = −cx²) and substitute or multiply term by term.
\[ \lim_{x \to 0^+}\left(\frac{\frac{x^{6}}{6} - \frac{x^{4}}{2} + x^{2} - 1 + e^{- x^{2}}}{x^{6}}\right) = 0 \]
These terms match f through x^6.✓ Proved

Answer \( - \frac{x^{6}}{6} + \frac{x^{4}}{2} - x^{2} + 1 + \cdots \)

Lines: 1 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	Not checked	—	a sentence; read, not computed
2	✓ 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	coefficients from derivatives at 0, and f − T is tiny at x = 0.01

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (misleading) — The stated answer lists the terms in descending order of power, which is non-standard for Maclaurin series and confusing. More critically, the solution text suggests substituting u = -cx^2 but does not specify c=1, and the equation check is for the remainder term rather than explicitly deriving the series terms, making the logical connection opaque.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (misleading) 2026-10-04 — The stated answer lists the terms in descending order of power, which is non-standard for Maclaurin series and confusing. More critically, the solution text suggests substituting u = -cx^2 but does not specify c=1, and the equation check is for the remainder term rather than explicitly deriving the series terms, making the logical connection opaque.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The stated answer lists the terms in descending order of power, which is non-standard for Maclaurin series and confusing. More critically, the solution does not explicitly derive the terms, relying on an unchecked limit equation that does not constitute a valid derivation step for a student.
gpt-oss:20b: pass 2026-10-04

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/maclaurin_substitution, checked 2026-10-04 with SymPy 1.14.0.