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Maclaurin series by substitution

Problem 7.275 · medium

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

Start from a known series (x cos(cx)) and substitute or multiply term by term.
\[ \lim_{x \to 0^+}\left(\frac{\frac{x^{7}}{720} - \frac{x^{5}}{24} + \frac{x^{3}}{2} + x \cos{\left(x \right)} - x}{x^{7}}\right) = 0 \]
These terms match f through x^7.✓ Proved

Answer \( - \frac{x^{7}}{720} + \frac{x^{5}}{24} - \frac{x^{3}}{2} + x + \cdots \)

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 lists terms in descending order of power (x^7, x^5, x^3, x), which is not the standard convention for Maclaurin series (ascending order). More critically, the signs are incorrect: the expansion of x*cos(x) is x - x^3/2 + x^5/24 - x^7/720 + ..., but the answer gives -x^7/720 + x^5/24 - x^3/2 + x. While the terms present are correct, the ordering is non-standard and potentially confusing, but the primary issue is that the solution text is vague ('substitute or multiply') and does not explicitly derive the terms, relying on a limit check that confirms the terms match but doesn't explain the derivation process required by 'Use a known Maclaurin series'. However, looking closely at the stated answer: -x^7/720 + x^5/24 - x^3/2 + x. This is actually x - x^3/2 + x^5/24 - x^7/720 written in reverse order. The signs are correct for the standard expansion. The main failure is that the solution sentence is too vague to be a proper derivation, and the final answer format is unconventional. But wait, is it an error? The problem asks for the first four nonzero terms. The answer provides them. The order is unusual but not mathematically wrong. However, the solution sentence 'Start from a known series (x cos(cx))' is misleading because the standard known series is for cos(x), not x*cos(cx). One starts with cos(x) and multiplies by x. Starting with 'x cos(cx)' implies a specific form that isn't the standard 'known series'. This is a pedagogical error/misleading statement.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The stated answer lists terms in descending order of power (x^7, x^5, x^3, x), which is not the standard convention for Maclaurin series (ascending order). More critically, the signs are incorrect: the expansion of x*cos(x) is x - x^3/2 + x^5/24 - x^7/720 + ..., but the answer gives -x^7/720 + x^5/24 - x^3/2 + x. While the terms present are correct, the ordering is non-standard and potentially confusing, but the primary issue is that the solution text is vague ('substitute or multiply') and does not explicitly derive the terms, relying on a limit check that confirms the terms match but doesn't explain the derivation process required by 'Use a known Maclaurin series'. However, looking closely at the stated answer: -x^7/720 + x^5/24 - x^3/2 + x. This is actually x - x^3/2 + x^5/24 - x^7/720 written in reverse order. The signs are correct for the standard expansion. The main failure is that the solution sentence is too vague to be a proper derivation, and the final answer format is unconventional. But wait, is it an error? The problem asks for the first four nonzero terms. The answer provides them. The order is unusual but not mathematically wrong. However, the solution sentence 'Start from a known series (x cos(cx))' is misleading because the standard known series is for cos(x), not x*cos(cx). One starts with cos(x) and multiplies by x. Starting with 'x cos(cx)' implies a specific form that isn't the standard 'known series'. This is a pedagogical error/misleading statement.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The stated answer is incorrect; it lists terms of degree 1, 3, 5, and 7, but the Maclaurin series for x cos(x) contains only odd powers starting with x (x - x^3/2 + x^5/24 - ...). The term -x^7/720 is the fourth nonzero term, but the solution presents them in descending order of magnitude or mixed order, and crucially, the coefficient of x^3 is -1/2, not -1/2 (wait, -x^3/2 is correct), but the term x^5/24 is correct, and -x^7/720 is correct. However, the standard form lists terms in ascending order of power. More importantly, the solution claims to find the 'first four nonzero terms'. The series is x - x^3/2 + x^5/24 - x^7/720 + ... The stated answer is -x^7/720 + x^5/24 - x^3/2 + x. This is the same set of terms, just reversed. Is that an error? Usually, series are written in ascending order. But the bigger issue is the method description. Step 1 says 'Start from a known series (x cos(cx))'. This is circular or confusing. One should start from cos(x) and multiply by x. Also, the equation check is for the difference between the polynomial and the function divided by x^7 going to 0, which confirms the polynomial matches up to x^6, but since the next term is x^7, it matches through x^7. The main error is likely the presentation order or the confusing setup sentence. Let's look closer. The problem asks for the first four nonzero terms. The answer provides them. The order is unconventional but mathematically equivalent as a sum. However, the sentence 'Start from a known series (x cos(cx))' is poor advice. One starts from cos(x). Also, the variable 'c' is introduced but not used. This is misleading. But is it an 'error'? The final answer is correct in content. Let's check the coefficients again. cos(x) = 1 - x^2/2 + x^4/24 - x^6/720 + ... x cos(x) = x - x^3/2 + x^5/24 - x^7/720 + ... The stated answer is -x^7/720 + x^5/24 - x^3/2 + x. This is correct. The sentence is just bad style/confusing. Wait, the prompt says 'find what is wrong'. If the answer is correct, is it a pass? The sentence 'Start from a known series (x cos(cx))' implies there is a known series for x cos(cx) which is not standard. Standard is cos(x). This is misleading. I will mark it as misleading because it suggests a non-standard starting point.
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.