Maclaurin series by substitution
Problem 7.354 · medium
Use a known Maclaurin series to find the first four nonzero terms of the Maclaurin series of \( \displaystyle f(x) = \frac{\sin{\left(3 x \right)}}{x} \).
- Start from a known series (sin u / x) and substitute or multiply term by term.
- \[ \lim_{x \to 0^+}\left(\frac{\frac{243 x^{6}}{560} - \frac{81 x^{4}}{40} + \frac{9 x^{2}}{2} - 3 + \frac{\sin{\left(3 x \right)}}{x}}{x^{6}}\right) = 0 \]These terms match f through x^6.✓ Proved
Answer \( - \frac{243 x^{6}}{560} + \frac{81 x^{4}}{40} - \frac{9 x^{2}}{2} + 3 + \cdots \)
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: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer has the wrong signs for all terms. The Maclaurin series for sin(3x)/x is 3 - (9/2)x^2 + (81/40)x^4 - (243/560)x^6 + ..., but the p
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer has the wrong signs for all terms. The Maclaurin series for sin(3x)/x is 3 - (9/2)x^2 + (81/40)x^4 - (243/560)x^6 + ..., but the pqwen3.6:27b-mlx: fail (error) 2026-10-06 — The stated answer is incorrect; the first term of the Maclaurin series for sin(3x)/x is 3, but the provided series starts with a negative x^6 term and lists terms in descending order of power, which is not a Maclaurin series. The solution fails to derive the correct series: 3 - (3^3/3!)x^2 + (3^5/5!)x^4 - ... = 3 - 9/2 x^2 + 27/8 x^4 - ...gpt-oss:20b: fail (misleading) 2026-10-06 — The solution’s first sentence vaguely references a known series but does not specify it, and the second sentence presents a limit test with incorrect signs for the proposed terms, misleading the reader about the correct Maclaurin expansion.
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.