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Maclaurin series by substitution

Problem 7.272 · medium

Use a known Maclaurin series to find the first four nonzero terms of the Maclaurin series of \( \displaystyle f(x) = \frac{1}{1 - x^{3}} \).

Start from a known series (geometric with u = x³) and substitute or multiply term by term.
\[ \lim_{x \to 0^+}\left(\frac{- x^{9} - x^{6} - x^{3} - 1 + \frac{1}{1 - x^{3}}}{x^{9}}\right) = 0 \]
These terms match f through x^9.✓ Proved

Answer \( x^{9} + x^{6} + x^{3} + 1 + \cdots \)

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: pass
qwen3.6:27b-mlx: fail (misleading) — The stated answer lists the terms in descending order of power (x^9, x^6, x^3, 1), which is non-standard for Maclaurin series and contradicts the standard convention of ascending powers. Additionally, the solution lacks the explicit derivation steps showing the substitution into the geometric series formula.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (misleading) 2026-10-04 — The stated answer lists the terms in descending order of power (x^9, x^6, x^3, 1), which is non-standard for Maclaurin series and contradicts the standard convention of ascending powers. Additionally, the solution lacks the explicit derivation steps showing the substitution into the geometric series formula.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The stated answer lists the terms in descending order of powers (x^9, x^6, x^3, 1), which is not the standard convention for Maclaurin series (ascending order: 1, x^3, x^6, x^9). While the terms are correct, the presentation is non-standard and potentially confusing for a student learning series expansions.
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.