Maclaurin series by substitution
Problem 7.270 · medium
Use a known Maclaurin series to find the first four nonzero terms of the Maclaurin series of \( \displaystyle f(x) = x^{2} \sin{\left(x \right)} \).
- Start from a known series (x² sin x) and substitute or multiply term by term.
- \[ \lim_{x \to 0^+}\left(\frac{\frac{x^{9}}{5040} - \frac{x^{7}}{120} + \frac{x^{5}}{6} - x^{3} + x^{2} \sin{\left(x \right)}}{x^{9}}\right) = 0 \]These terms match f through x^9.✓ Proved
Answer \( - \frac{x^{9}}{5040} + \frac{x^{7}}{120} - \frac{x^{5}}{6} + x^{3} + \cdots \)
Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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: fail (misleading) — The solution incorrectly states that the known series is "x² sin x"; the Maclaurin series should be started from sin x, then multiplied by x². This could mislead a student into thinking the series for x² sin x is known directly.qwen3.6:27b-mlx: fail (error) — The stated answer lists terms with incorrect signs (e.g., -x^5/6 instead of x^5/6) and in descending order of powers, which is not the standard form for a Maclaurin series. The solution text is vague and does not correct these errors.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The stated answer lists terms with incorrect signs (e.g., -x^5/6 instead of x^5/6) and in descending order of powers, which is not the standard form for a Maclaurin series. The solution text is vague and does not correct these errors.gpt-oss:20b: fail (misleading) 2026-10-04 — The solution incorrectly states that the known series is "x² sin x"; the Maclaurin series should be started from sin x, then multiplied by x². This could mislead a student into thinking the series for x² sin x is known directly.qwen3.6:27b-mlx: fail (error) 2026-10-04 — The stated answer is incorrect because it includes a term (-x^9/5040) that does not exist in the Maclaurin series of x^2 sin(x), which only contains odd powers starting from x^3. The solution fails to derive the correct series and instead presents a wrong result.gpt-oss:20b: inconclusive 2026-10-04 — reviewer response could not be parsed: {"verdict":"fail","severity":"misleading","notes":"The first sentence incorrectly claims to start from a known series for \(x^{2}\sin x\); the correct approach is to use the Maclaurin series for \(\si
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.