Maclaurin series by substitution
Problem 7.361 · medium
Use a known Maclaurin series to find the first four nonzero terms of the Maclaurin series of \( \displaystyle f(x) = x e^{x} \).
- Start from a known series (x times e^u, u = cx) and substitute or multiply term by term.
- \[ \lim_{x \to 0^+}\left(\frac{- \frac{x^{4}}{6} - \frac{x^{3}}{2} - x^{2} + x e^{x} - x}{x^{4}}\right) = 0 \]These terms match f through x^4.✓ Proved
Answer \( \frac{x^{4}}{6} + \frac{x^{3}}{2} + x^{2} + x + \cdots \)
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 is incorrect; the first four nonzero terms of x*e^x are x + x^2/2 + x^3/6 + x^4/24, but the solution lists x + x^2 + x^3/2 + x^4/6. The coefficients are wrong (missing the division by n!). The equation check is misleading because it verifies a specific linear combination that happens to vanish to order 4, but does not prove the coefficients in the stated answer are correct for the function.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The stated answer is incorrect; the first four nonzero terms of x*e^x are x + x^2/2 + x^3/6 + x^4/24, but the solution lists x + x^2 + x^3/2 + x^4/6. The coefficients are wrong (missing the division by n!). The equation check is misleading because it verifies a specific linear combination that happens to vanish to order 4, but does not prove the coefficients in the stated answer are correct for the function.qwen3.6:27b-mlx: fail (error) 2026-10-06 — The stated answer lists the terms in descending order of power ($x^4, x^3, x^2, x$), which is non-standard for Maclaurin series and confusing. More critically, the solution's verification equation checks the expansion of $x e^{-x}$ (terms are $x - x^2 + x^3/2 - x^4/6$) rather than $x e^x$ (terms are $x + x^2 + x^3/2 + x^4/6$). The signs in the verification limit are wrong for the function $f(x) = x e^x$.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.