Maclaurin series by substitution
Problem 7.357 · medium
Use a known Maclaurin series to find the first four nonzero terms of the Maclaurin series of \( \displaystyle f(x) = \operatorname{atan}{\left(x^{2} \right)} \).
- Start from a known series (arctan u with u = x²) and substitute or multiply term by term.
- \[ \lim_{x \to 0^+}\left(\frac{\frac{x^{14}}{7} - \frac{x^{10}}{5} + \frac{x^{6}}{3} - x^{2} + \operatorname{atan}{\left(x^{2} \right)}}{x^{14}}\right) = 0 \]These terms match f through x^14.✓ Proved
Answer \( - \frac{x^{14}}{7} + \frac{x^{10}}{5} - \frac{x^{6}}{3} + x^{2} + \cdots \)
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 the Maclaurin series of atan(x^2). The series for atan(u) is u - u^3/3 + u^5/5 - ..., so atan(x^2) = x^2 -
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 the Maclaurin series of atan(x^2). The series for atan(u) is u - u^3/3 + u^5/5 - ..., so atan(x^2) = x^2 -qwen3.6:27b-mlx: fail (error) 2026-10-06 — The stated answer is incorrect because it lists terms with negative coefficients for the higher powers, whereas the Maclaurin series for arctan(x^2) has alternating signs starting with a positive x^2 term (x^2 - x^6/3 + x^10/5 - x^14/7 + ...). The provided solution claims the terms match, but the signs in the stated answer are flipped for the x^6, x^10, and x^14 terms relative to the standard expansion.gpt-oss:20b: pass 2026-10-06
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.