Maclaurin series by substitution
Problem 7.356 · medium
Use a known Maclaurin series to find the first four nonzero terms of the Maclaurin series of \( \displaystyle f(x) = e^{- 2 x^{2}} \).
- Start from a known series (e^u with u = −cx²) and substitute or multiply term by term.
- \[ \lim_{x \to 0^+}\left(\frac{\frac{4 x^{6}}{3} - 2 x^{4} + 2 x^{2} - 1 + e^{- 2 x^{2}}}{x^{6}}\right) = 0 \]These terms match f through x^6.✓ Proved
Answer \( - \frac{4 x^{6}}{3} + 2 x^{4} - 2 x^{2} + 1 + \cdots \)
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: passqwen3.6:27b-mlx: fail (error) — The stated answer has the wrong signs for the terms. The Maclaurin series for e^u is 1 + u + u^2/2 + ..., so for u = -2x^2, the terms are 1 - 2x^2 + 2x^4 - 4x^6/3 + .... The provided answer lists 1 - 2x^2 + 2x^4 - 4x^6/3 but writes it as -4x^6/3 + 2x^4 - 2x^2 + 1, which is actually correct in value but the prompt's stated answer string is '- 4 x^{6}/3 + 2 x^{4} - 2 x^{2} + 1'. Wait, let me re-read carefully. The stated answer is '- 4 x^{6}/3 + 2 x^{4} - 2 x^{2} + 1'. This is mathematically equivalent to 1 - 2x^2 + 2x^4 - 4x^6/3. The signs are correct. Why did I think they were wrong? Let's check the equation line. The equation checks `Limit((4*x**6/3 - 2*x**4 + 2*x**2 - 1 + exp(-2*x**2))/x**6, x, 0, dir='+') = 0`. This implies `exp(-2*x**2) - (1 - 2x^2 + 2x^4 - 4x^6/3)` is O(x^6). Wait, the numerator is `4*x**6/3 - 2*x**4 + 2*x**2 - 1 + exp(-2*x**2)`. If the limit is 0, then `exp(-2*x**2) ≈ 1 - 2x^2 + 2x^4 - 4x^6/3`. The stated answer is `- 4 x^{6}/3 + 2 x^{4} - 2 x^{2} + 1`. This matches. So the math is correct. Is there a subtle error? The problem asks for the first four nonzero terms. The solution provides them. The sentence is vague but not wrong. The equation confirms the result. It seems correct. Let me double check the 'adversarial' nature. Is the order of terms an issue? Usually series are written in ascending powers of x. The stated answer is in descending powers. This is unconventional but not 'wrong'. However, the sentence 'Start from a known series... and substitute' is a bit hand-wavy but acceptable. Let's look closer at the equation. `4*x**6/3 - 2*x**4 + 2*x**2 - 1`. This is `-(1 - 2x^2 + 2x^4 - 4x^6/3)`. So `exp(...) - (1 - 2x^2 + 2x^4 - 4x^6/3)` has limit 0. This confirms the series expansion is `1 - 2x^2 + 2x^4 - 4x^6/3`. The stated answer is `- 4 x^{6}/3 + 2 x^{4} - 2 x^{2} + 1`. This is the same polynomial. Verdict: pass.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The stated answer has the wrong signs for the terms. The Maclaurin series for e^u is 1 + u + u^2/2 + ..., so for u = -2x^2, the terms are 1 - 2x^2 + 2x^4 - 4x^6/3 + .... The provided answer lists 1 - 2x^2 + 2x^4 - 4x^6/3 but writes it as -4x^6/3 + 2x^4 - 2x^2 + 1, which is actually correct in value but the prompt's stated answer string is '- 4 x^{6}/3 + 2 x^{4} - 2 x^{2} + 1'. Wait, let me re-read carefully. The stated answer is '- 4 x^{6}/3 + 2 x^{4} - 2 x^{2} + 1'. This is mathematically equivalent to 1 - 2x^2 + 2x^4 - 4x^6/3. The signs are correct. Why did I think they were wrong? Let's check the equation line. The equation checks `Limit((4*x**6/3 - 2*x**4 + 2*x**2 - 1 + exp(-2*x**2))/x**6, x, 0, dir='+') = 0`. This implies `exp(-2*x**2) - (1 - 2x^2 + 2x^4 - 4x^6/3)` is O(x^6). Wait, the numerator is `4*x**6/3 - 2*x**4 + 2*x**2 - 1 + exp(-2*x**2)`. If the limit is 0, then `exp(-2*x**2) ≈ 1 - 2x^2 + 2x^4 - 4x^6/3`. The stated answer is `- 4 x^{6}/3 + 2 x^{4} - 2 x^{2} + 1`. This matches. So the math is correct. Is there a subtle error? The problem asks for the first four nonzero terms. The solution provides them. The sentence is vague but not wrong. The equation confirms the result. It seems correct. Let me double check the 'adversarial' nature. Is the order of terms an issue? Usually series are written in ascending powers of x. The stated answer is in descending powers. This is unconventional but not 'wrong'. However, the sentence 'Start from a known series... and substitute' is a bit hand-wavy but acceptable. Let's look closer at the equation. `4*x**6/3 - 2*x**4 + 2*x**2 - 1`. This is `-(1 - 2x^2 + 2x^4 - 4x^6/3)`. So `exp(...) - (1 - 2x^2 + 2x^4 - 4x^6/3)` has limit 0. This confirms the series expansion is `1 - 2x^2 + 2x^4 - 4x^6/3`. The stated answer is `- 4 x^{6}/3 + 2 x^{4} - 2 x^{2} + 1`. This is the same polynomial. Verdict: pass.qwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer lists the terms in descending order of power, which is non-standard for Maclaurin series, and more critically, the signs are incorgpt-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.