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Maclaurin series by substitution

Problem 7.358 · medium

Use a known Maclaurin series to find the first four nonzero terms of the Maclaurin series of \( \displaystyle f(x) = x \cos{\left(3 x \right)} \).

Start from a known series (x cos(cx)) and substitute or multiply term by term.
\[ \lim_{x \to 0^+}\left(\frac{\frac{81 x^{7}}{80} - \frac{27 x^{5}}{8} + \frac{9 x^{3}}{2} + x \cos{\left(3 x \right)} - x}{x^{7}}\right) = 0 \]
These terms match f through x^7.✓ Proved

Answer \( - \frac{81 x^{7}}{80} + \frac{27 x^{5}}{8} - \frac{9 x^{3}}{2} + x + \cdots \)

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 is incorrect; the coefficient of x^5 should be 27/8, but the coefficient of x^7 is -81/160, not -81/80. The solution claims the terms match through x^7, which is false.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: fail (error) 2026-10-06 — The stated answer is incorrect; the coefficient of x^5 should be 27/8, but the coefficient of x^7 is -81/160, not -81/80. The solution claims the terms match through x^7, which is false.
qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the method (using the known Maclaurin series for cosine and multiplying by x) and the final answer matches the derived series terms. The algebraic check confirms the coefficients are correct.
gpt-oss:20b: fail (misleading) 2026-10-06 — The solution claims to use a known series but never actually derives or displays the Maclaurin coefficients; the limit test is insufficient to justify the stated terms, so a student would be misled into thinking the series was correctly obtained.

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.