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Maclaurin series by substitution

Problem 7.355 · medium

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

Start from a known series (x times e^u, u = cx) and substitute or multiply term by term.
\[ \lim_{x \to 0^+}\left(\frac{- \frac{4 x^{4}}{3} - 2 x^{3} - 2 x^{2} + x e^{2 x} - x}{x^{4}}\right) = 0 \]
These terms match f through x^4.✓ Proved

Answer $ \frac{4 x^{4}}{3} + 2 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: pass
qwen3.6:27b-mlx: fail (error) — 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 coefficient for the x^4 term is incorrect: the expansion of x*e^(2x) is x + 2x^2 + 4/3 x^3 + 2/3 x^4 + ..., so the x^4 term should be 2/3 x^4, not 4/3 x^4. The provided limit check in the solution is also algebraically inconsistent with the stated answer (it checks for -4/3, implying a sign error or mismatch with the positive answer provided).

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-06
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 coefficient for the x^4 term is incorrect: the expansion of x*e^(2x) is x + 2x^2 + 4/3 x^3 + 2/3 x^4 + ..., so the x^4 term should be 2/3 x^4, not 4/3 x^4. The provided limit check in the solution is also algebraically inconsistent with the stated answer (it checks for -4/3, implying a sign error or mismatch with the positive answer provided).
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 misleading regarding the 'first four nonzero terms' (which are $x, 2x^2, 4x^3, 4x^4/3$). Furthermore, the coefficient of $x^4$ in the stated answer is $4/3$, but the correct coefficient is $4/24 = 1/6$. The limit check in line 2 uses $-4/3$, confirming the solution is checking against the incorrect stated answer rather than the true function.
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.