Maclaurin series by substitution
Problem 7.278 · medium
Use a known Maclaurin series to find the first four nonzero terms of the Maclaurin series of \( \displaystyle f(x) = \cos{\left(x^{2} \right)} \).
- Start from a known series (cos u with u = x²) and substitute or multiply term by term.
- \[ \lim_{x \to 0^+}\left(\frac{\frac{x^{12}}{720} - \frac{x^{8}}{24} + \frac{x^{4}}{2} + \cos{\left(x^{2} \right)} - 1}{x^{12}}\right) = 0 \]These terms match f through x^12.✓ Proved
Answer \( - \frac{x^{12}}{720} + \frac{x^{8}}{24} - \frac{x^{4}}{2} + 1 + \cdots \)
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: fail (misleading) — The limit expression uses the wrong signs for the series terms; it should be (-x**12/720 + x**8/24 - x**4/2 + cos(x**2) - 1)/x**12 → 0, not the positive signs shown. This would mislead a student about the correct Maclaurin expansion.qwen3.6:27b-mlx: pass — The solution correctly identifies the method (substitution into the Maclaurin series for cosine) and the verification step confirms the first four nonzero terms are correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the method (substitution into the Maclaurin series for cosine) and the verification step confirms the first four nonzero terms are correct.gpt-oss:20b: fail (misleading) 2026-10-04 — The limit expression uses the wrong signs for the series terms; it should be (-x**12/720 + x**8/24 - x**4/2 + cos(x**2) - 1)/x**12 → 0, not the positive signs shown. This would mislead a student about the correct Maclaurin expansion.qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the method of substituting u = x^2 into the Maclaurin series for cos(u). The resulting terms match the stated answer, and the limit check confirms the correctness of the expansion up to the specified order.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.