Home›Calculus 2›Maclaurin series by substitution›Problem 7.362

Maclaurin series by substitution

Problem 7.362 · medium

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

Start from a known series (x times e^u, u = cx) and substitute or multiply term by term.
\[ \lim_{x \to 0^+}\left(\frac{- \frac{9 x^{4}}{2} - \frac{9 x^{3}}{2} - 3 x^{2} + x e^{3 x} - x}{x^{4}}\right) = 0 \]
These terms match f through x^4.✓ Proved

Answer \( \frac{9 x^{4}}{2} + \frac{9 x^{3}}{2} + 3 x^{2} + x + \cdots \)

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 (error) — The stated answer lists the terms in descending order of power (x^4, x^3, x^2, x), which is non-standard for a Maclaurin series and contradicts the standard convention of ascending powers. Furthermore, the coefficient for the x^4 term is incorrect: the expansion of x*e^(3x) is x + 3x^2 + 9/2 x^3 + 27/8 x^4 + ..., so the x^4 term should be 27/8 x^4, not 9/2 x^4.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: fail (error) 2026-10-06 — The stated answer lists the terms in descending order of power (x^4, x^3, x^2, x), which is non-standard for a Maclaurin series and contradicts the standard convention of ascending powers. Furthermore, the coefficient for the x^4 term is incorrect: the expansion of x*e^(3x) is x + 3x^2 + 9/2 x^3 + 27/8 x^4 + ..., so the x^4 term should be 27/8 x^4, not 9/2 x^4.
qwen3.6:27b-mlx: fail (error) 2026-10-06 — The stated answer is incorrect; the coefficient of x^3 is 9/2, but it should be 9/2! = 9/2 (wait, 3^3/3! = 27/6 = 9/2, so that term is correct? Let's recheck. e^{3x} = 1 + 3x + 9x^2/2 + 27x^3/6 + ... = 1 + 3x + 9/2 x^2 + 9/2 x^3 + ... Multiplying by x gives x + 3x^2 + 9/2 x^3 + 9/2 x^4. The stated answer has 9/2 x^4 + 9/2 x^3 + 3x^2 + x. This matches. Wait, let me look closer at the equation line. The equation checks `x*exp(3*x) - (x + 3x^2 + 9/2 x^3 + 9/2 x^4)`. The limit is 0. So the terms are correct. Why did I think it was wrong? Ah, the prompt says 'first four nonzero terms'. The solution lists them. Is there an error? Let's re-read the solution text. 'Start from a known series... substitute or multiply'. This is vague but not wrong. The equation confirms the terms. Is the order wrong? Usually series are written in ascending powers. The answer is written in descending powers. This is unusual but not mathematically wrong. However, looking at the equation: `Limit((-9*x**4/2 - 9*x**3/2 - 3*x**2 + x*exp(3*x) - x)/x**4, x, 0, dir='+') = 0`. This implies the terms subtracted are `x + 3x^2 + 9/2 x^3 + 9/2 x^4`. The stated answer is `9/2 x^4 + 9/2 x^3 + 3x^2 + x`. These are the same terms. So the math is correct. Is there a subtle error? The problem asks for the Maclaurin series. The solution provides the terms. The sentence is just a strategy. The equation verifies it. It seems correct. Wait, let me check the coefficients again. f(x) = x * sum( (3x)^n / n! ) = sum( 3^n x^{n+1} / n! ). n=0: 3^0 x^1 / 0! = x. n=1: 3^1 x^2 / 1! = 3x^2. n=2: 3^2 x^3 / 2! = 9/2 x^3. n=3: 3^3 x^4 / 3! = 27/6 x^4 = 9/2 x^4. The terms are x, 3x^2, 9/2 x^3, 9/2 x^4. The stated answer lists them. The equation verifies them. Why would this fail? Maybe the 'sentence' is too vague? 'Start from a known series (x times e^u, u = cx) and substitute or multiply term by term.' This is a valid method. Is there a convention issue? Usually, series are written in increasing order of powers. Writing them in decreasing order is 'style' but not 'error'. However, the prompt asks for 'first four nonzero terms'. The order doesn't matter for the set of terms, but conventionally one lists them starting from the lowest power. If I mark it 'style', that might be appropriate. But let's look for a harder error. Did the user make a mistake in the equation check? `x*exp(3*x) - x - 3x^2 - 9/2 x^3 - 9/2 x^4`. The limit of this divided by x^4 is 0. This is correct. So the terms are correct. Is there any other interpretation? Maybe the 'sentence' implies a substitution method that is harder than necessary? No, multiplying by x is standard. I will mark it as pass, but wait, the instructions say 'find what is wrong'. If it's correct, I say pass. Let me double check the 'stated answer' vs 'solution'. The solution *is* the derivation. The stated answer is the result. The result is correct. The derivation is correct. I will pass it.
gpt-oss:20b: fail (error) 2026-10-06 — The solution does not actually produce the first four non‑zero terms of the Maclaurin series. It merely states a limit test without showing the expansion or verifying that the coefficients are correct. The setup is incomplete and would not teach a student how to obtain the required terms.

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.