Limits using Taylor series
Problem 7.673 · medium
Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{2 x - 2 \ln{\left(x + 1 \right)}}{x^{2}} \).
- Near 0, -x + log(x + 1) = x**3/3 - x**2/2 + ⋯ (its Maclaurin series).
- \[ \lim_{x \to 0}\left(\frac{- \frac{x^{3}}{3} + \frac{x^{2}}{2} - x + \ln{\left(x + 1 \right)}}{x^{3}}\right) = 0 \]The polynomial matches the numerator to that order.✓ Proved
- \[ \frac{- \frac{2 x^{3}}{3} + x^{2}}{x^{2}} = 1 - \frac{2 x}{3} \]Divide through by the denominator.✓ Proved
- \[ \lim_{x \to 0}\left(\frac{2 x - 2 \ln{\left(x + 1 \right)}}{x^{2}}\right) = 1 \]Let x → 0: only the constant term survives.✓ Proved
Answer \( 1 \)
Lines: 3 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. How we verify
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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ 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 | the expression at x = ±10⁻⁸ in 50-digit arithmetic |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Line 1 contains a sign error in the Maclaurin series expansion: it claims -x + log(x+1) = x^3/3 - x^2/2 + ..., but the correct expansion is -x^2/2 +
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Line 1 contains a sign error in the Maclaurin series expansion: it claims -x + log(x+1) = x^3/3 - x^2/2 + ..., but the correct expansion is -x^2/2 +gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution claims the Maclaurin series for -x + log(x + 1) is x**3/3 - x**2/2 + ..., but the actual series is -x**2/2 + x**3/3 - ... (the x**3 term has the wrong sign in the text, though the subsequent algebraic substitution in step 3 uses the correct term x**3/3, creating a contradiction between the stated series and the calculation). Furthermore, step 2 is irrelevant and confusingly presented.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The Maclaurin series expansion in step 1 is incorrect; the expansion of log(x+1) is x - x^2/2 + x^3/3 - ..., so -x + log(x+1) equals -x^2/2 + x^3/3 - ..., not x^3/3 - x^2/2 + ... (the signs of the terms are swapped relative to the standard expansion, though the specific combination -x^2/2 + x^3/3 is actually correct for -x + log(x+1), the text claims -x + log(x+1) = x^3/3 - x^2/2 + ... which is numerically correct but the derivation logic in step 2 is confused. More critically, step 2 calculates a limit of order 3 which is irrelevant to the problem, and step 3 divides by x^2 but uses the wrong numerator expression from step 1's implied structure. The primary error is that step 1 states '-x + log(x+1) = x^3/3 - x^2/2 + ...' which is actually correct (-x + (x - x^2/2 + x^3/3) = -x^2/2 + x^3/3). However, step 2 introduces a completely different limit expression involving division by x^3 which is not part of the problem statement and serves no purpose. Step 3 then arbitrarily picks terms from the series to divide by x^2. The logic is disjointed and the intermediate steps do not follow from the problem statement.gpt-oss:20b: pass 2026-10-15
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/taylor_limit, checked 2026-10-18 with SymPy 1.14.0.