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Maclaurin series by substitution

Problem 7.276 · medium

Use a known Maclaurin series to find the first four nonzero terms of the Maclaurin series of \( \displaystyle f(x) = \ln{\left(2 x + 1 \right)} \).

Start from a known series (ln(1 + u), u = cx) and substitute or multiply term by term.
Reviewed
\[ \lim_{x \to 0^+}\left(\frac{4 x^{4} - \frac{8 x^{3}}{3} + 2 x^{2} - 2 x + \ln{\left(2 x + 1 \right)}}{x^{4}}\right) = 0 \]
These terms match f through x^4.✓ Proved

Answer \( - 4 x^{4} + \frac{8 x^{3}}{3} - 2 x^{2} + 2 x + \cdots \)

Lines: 1 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly identifies the method (substitution into the standard ln(1+u) series) and the final answer is algebraically correct and matches the problem statement.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the method (substitution into the standard ln(1+u) series) and the final answer is algebraically correct and matches the problem statement.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: inconclusive 2026-10-04 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer has incorrect signs for the terms. The Maclaurin series for ln(1+u) is u - u^2/2 + u^3/3 - u^4/4 + ... Substituting u=2x giv
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.