Maclaurin series by substitution
Problem 7.359 · medium
Use a known Maclaurin series to find the first four nonzero terms of the Maclaurin series of \( \displaystyle f(x) = \cos{\left(2 x^{2} \right)} \).
- Start from a known series (cos u with u = x²) and substitute or multiply term by term.
- \[ \lim_{x \to 0^+}\left(\frac{\frac{4 x^{12}}{45} - \frac{2 x^{8}}{3} + 2 x^{4} + \cos{\left(2 x^{2} \right)} - 1}{x^{12}}\right) = 0 \]These terms match f through x^12.✓ Proved
Answer \( - \frac{4 x^{12}}{45} + \frac{2 x^{8}}{3} - 2 x^{4} + 1 + \cdots \)
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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution incorrectly states to use the series for \(\cos u\) with \(u=x^2\); the function is \(\cos(2x^2)\), so the correct substitution is \(u=2x^2\)qwen3.6:27b-mlx: fail (error) — The setup sentence incorrectly states to substitute u = x^2, whereas the problem requires u = 2x^2. Although the final equation is algebraically correct, the described method does not match the problem statement.
Every verdict on record (4)
gpt-oss:20b: inconclusive 2026-10-06 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution incorrectly states to use the series for \(\cos u\) with \(u=x^2\); the function is \(\cos(2x^2)\), so the correct substitution is \(u=2x^2\)qwen3.6:27b-mlx: fail (error) 2026-10-06 — The setup sentence incorrectly states to substitute u = x^2, whereas the problem requires u = 2x^2. Although the final equation is algebraically correct, the described method does not match the problem statement.qwen3.6:27b-mlx: fail (error) 2026-10-06 — The setup sentence incorrectly states to substitute u = x^2, but the problem requires u = 2x^2. This is a fundamental error in the method description, even though the final equation happens to be correct.gpt-oss:20b: inconclusive 2026-10-06 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution incorrectly starts from the Maclaurin series for \(\cos u\) with \(u=x^2\) instead of the required \(u=2x^2\). Consequently the signs of the
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.