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Maclaurin series by substitution

Problem 7.353 · medium

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

Start from a known series (x cos(cx)) and substitute or multiply term by term.
\[ \lim_{x \to 0^+}\left(\frac{\frac{4 x^{7}}{45} - \frac{2 x^{5}}{3} + 2 x^{3} + x \cos{\left(2 x \right)} - x}{x^{7}}\right) = 0 \]
These terms match f through x^7.✓ Proved

Answer $ - \frac{4 x^{7}}{45} + \frac{2 x^{5}}{3} - 2 x^{3} + 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 term $-\frac{4x^7}{45}$ does not appear in the Maclaurin series of $x \cos(2x)$, which contains only odd powers $x, x^3, x^5, x^7$ with coefficients derived from $\cos(2x)$. Specifically, the $x^7$ term should be $-\frac{2}{7!}x^7 = -\frac{1}{1260}x^7$, not $-\frac{4}{45}x^7$. The solution's equation line claims these terms match, but they do not.

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 term $-\frac{4x^7}{45}$ does not appear in the Maclaurin series of $x \cos(2x)$, which contains only odd powers $x, x^3, x^5, x^7$ with coefficients derived from $\cos(2x)$. Specifically, the $x^7$ term should be $-\frac{2}{7!}x^7 = -\frac{1}{1260}x^7$, not $-\frac{4}{45}x^7$. The solution's equation line claims these terms match, but they do not.
qwen3.6:27b-mlx: fail (error) 2026-10-06 — The stated answer is incorrect; the first four nonzero terms of x cos(2x) are x - 2x^3 + 2x^5/3 - 4x^7/45. The solution claims the answer is -4x^7/45 + 2x^5/3 - 2x^3 + x, which is the same polynomial but listed in descending order of power, which is not the standard form for a Maclaurin series (ascending order). More critically, the 'unchecked' equation in line 2 verifies that the difference between the function and the polynomial is O(x^7), but it does not verify that the coefficients are correct for the *first four nonzero terms* specifically, nor does it address the ordering issue. However, looking closely at the stated answer: `- 4 x^{7}/45 + 2 x^{5}/3 - 2 x^{3} + x`. This is mathematically equivalent to `x - 2x^3 + 2/3 x^5 - 4/45 x^7`. The problem asks for the 'first four nonzero terms'. Usually, this implies listing them in increasing order of powers. The provided answer lists them in decreasing order. While mathematically the sum is the same, the convention for 'Maclaurin series' is ascending powers. Is this an error or style? The prompt asks to find the terms. The terms are correct. The order is non-standard. Let's look at the solution steps. Step 1 is vague. Step 2 is a check. The main issue is whether the answer format is acceptable. Usually, 'first four nonzero terms' implies `a_1 x + a_3 x^3 + ...`. The answer provided is `... + x`. This is just a reordering. However, often automated graders or strict conventions require ascending order. Let's look for a harder error. Wait, the coefficient of x^7 in cos(2x) is (-1)^3 (2x)^6 / 6! ? No. cos(u) = 1 - u^2/2! + u^4/4! - u^6/6! + ... u=2x. cos(2x) = 1 - 2x^2 + 2/3 x^4 - 4/45 x^6 + ... Multiply by x: x - 2x^3 + 2/3 x^5 - 4/45 x^7 + ... The stated answer is `- 4 x^{7}/45 + 2 x^{5}/3 - 2 x^{3} + x`. This is exactly the same terms. The order is descending. This is technically a 'style' issue if the content is correct. But wait, is there a sign error? No. Is there a coefficient error? No. The solution is 'sound' in terms of math, but the presentation is non-standard. However, the prompt says 'find the first four nonzero terms'. It doesn't explicitly say 'in ascending order'. But it is the universal convention. Let's look at the solution text again. 'Start from a known series (x cos(cx))'. This is not a standard 'known series'. One starts from cos(x) or cos(2x). Starting from 'x cos(cx)' is weird phrasing. But the main thing is the answer. If I mark it 'pass', I accept descending order. If I mark it 'fail', I enforce ascending order. Given 'adversarial' review, enforcing standard convention is appropriate. Also, the solution provides almost no work, just a limit check. The limit check `Limit((...)/x**7, x, 0) = 0` proves the error is O(x^7), meaning the terms up to x^6 are correct. It does not prove the x^7 term is correct (it would need to check x^8 or just the coefficient). Actually, if the limit is 0, it means the polynomial matches up to x^6. The x^7 term in the polynomial is -4/45. The x^7 term in the function is -4/45. The difference is 0 at order 7? No, the difference is O(x^8). The limit being 0 confirms the coefficients for x, x^3, x^5, x^7 are correct? No. If P(x) is the polynomial, f(x) - P(x) = O(x^8). Then (f-P)/x^7 -> 0. Yes. So the check is valid for the coefficients. The only issue is the ordering. I will mark it as style/fail for non-standard ordering.
gpt-oss:20b: fail (misleading) 2026-10-06 — The sentence claims the series terms match f through x^7, but the limit expression uses the wrong signs for the x^7, x^5, and x^3 terms, so a student would learn an incorrect series.

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.