Maclaurin series by substitution
Problem 7.271 · medium
Use a known Maclaurin series to find the first four nonzero terms of the Maclaurin series of \( \displaystyle f(x) = \frac{\sin{\left(x \right)}}{x} \).
- Start from a known series (sin u / x) and substitute or multiply term by term.
- \[ \lim_{x \to 0^+}\left(\frac{\frac{x^{6}}{5040} - \frac{x^{4}}{120} + \frac{x^{2}}{6} - 1 + \frac{\sin{\left(x \right)}}{x}}{x^{6}}\right) = 0 \]These terms match f through x^6.✓ Proved
Answer \( - \frac{x^{6}}{5040} + \frac{x^{4}}{120} - \frac{x^{2}}{6} + 1 + \cdots \)
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 (error) — The solution never actually writes down the Maclaurin series for sin x/x; it only references a vague “known series” and then checks a limit. The reasoning is incomplete and does not demonstrate how the first four non‑zero terms are obtained, so the setup and conclusion are incorrect.qwen3.6:27b-mlx: fail (error) — The stated answer lists terms in descending order of power, which is not the standard form for a Maclaurin series (ascending order). More critically, the solution text is vague and does not explicitly derive the terms from the known series for sin(x), relying instead on a limit check that confirms the result but does not demonstrate the method requested.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The stated answer lists terms in descending order of power, which is not the standard form for a Maclaurin series (ascending order). More critically, the solution text is vague and does not explicitly derive the terms from the known series for sin(x), relying instead on a limit check that confirms the result but does not demonstrate the method requested.gpt-oss:20b: fail (error) 2026-10-04 — The solution never actually writes down the Maclaurin series for sin x/x; it only references a vague “known series” and then checks a limit. The reasoning is incomplete and does not demonstrate how the first four non‑zero terms are obtained, so the setup and conclusion are incorrect.qwen3.6:27b-mlx: fail (error) 2026-10-04 — The stated answer lists the terms in descending order of power, which is not the standard convention for Maclaurin series (ascending order). More importantly, the solution text is vague and does not explicitly state the known series for sin(x) or the division process, making it an incomplete explanation.gpt-oss:20b: fail (error) 2026-10-04 — The solution does not actually produce the first four non‑zero terms of the Maclaurin series; it only states a limit test without showing the series itself. The setup sentence is vague and the conclusion is missing, so the answer is incomplete and incorrect.
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.