Maclaurin series by substitution
Problem 7.360 · medium
Use a known Maclaurin series to find the first four nonzero terms of the Maclaurin series of \( \displaystyle f(x) = e^{- 3 x^{2}} \).
- Start from a known series (e^u with u = −cx²) and substitute or multiply term by term.
- \[ \lim_{x \to 0^+}\left(\frac{\frac{9 x^{6}}{2} - \frac{9 x^{4}}{2} + 3 x^{2} - 1 + e^{- 3 x^{2}}}{x^{6}}\right) = 0 \]These terms match f through x^6.✓ Proved
Answer \( - \frac{9 x^{6}}{2} + \frac{9 x^{4}}{2} - 3 x^{2} + 1 + \cdots \)
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: passqwen3.6:27b-mlx: fail (error) — The stated answer has the wrong signs for all terms except the constant term. The Maclaurin series for e^(-3x^2) is 1 - 3x^2 + 9x^4/2 - 9x^6/2 + ..., but the answer lists +9x^4/2 and -9x^6/2. The provided limit check is also incorrect because it subtracts the wrong polynomial (with flipped signs) from the function, resulting in a non-zero limit or a mismatch that is not properly verified.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The stated answer has the wrong signs for all terms except the constant term. The Maclaurin series for e^(-3x^2) is 1 - 3x^2 + 9x^4/2 - 9x^6/2 + ..., but the answer lists +9x^4/2 and -9x^6/2. The provided limit check is also incorrect because it subtracts the wrong polynomial (with flipped signs) from the function, resulting in a non-zero limit or a mismatch that is not properly verified.qwen3.6:27b-mlx: fail (misleading) 2026-10-06 — The stated answer lists the terms in descending order of powers, which is non-standard for Maclaurin series and confusing. More importantly, the solution text is vague ('substitute or multiply') and does not explicitly show the substitution into the standard e^x series, which is the core of the problem.gpt-oss:20b: pass 2026-10-06
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 2026-10-06 with SymPy 1.14.0.