Maclaurin series by substitution
Problem 7.269 · medium
Use a known Maclaurin series to find the first four nonzero terms of the Maclaurin series of \( \displaystyle f(x) = - x + e^{x} - 1 \).
- Start from a known series (eˣ minus its first terms) and substitute or multiply term by term.
- \[ \lim_{x \to 0^+}\left(\frac{- \frac{x^{5}}{120} - \frac{x^{4}}{24} - \frac{x^{3}}{6} - \frac{x^{2}}{2} - x + e^{x} - 1}{x^{5}}\right) = 0 \]These terms match f through x^5.✓ Proved
Answer \( \frac{x^{5}}{120} + \frac{x^{4}}{24} + \frac{x^{3}}{6} + \frac{x^{2}}{2} + \cdots \)
Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 (style) — [domain objection, downgraded to style] The stated answer lists terms in descending order of power (x^5, x^4, ...), which is not the standard convention for Maclaurin series (ascending order). More critically, the stated answer includes x^2, x^3, x^4, and x^5, but the function f(x) = e^x - x - 1 has a nonzero x^2 term (1/2 x^2). The 'first four nonzero terms' should be 1/2 x^2 + 1/6 x^3 + 1/24 x^4 + 1/120 x^5. The provided answer is just a permutation of these, but typically 'first four' implies starting from the lowest degree. However, the main issue is that the solution text is vague ('substitute or multiply') and the equation check is for the limit of the remainder, which is a valid check but doesn't explain the derivation. The biggest error is that the stated answer is technically correct as a set of terms, but the ordering is non-standard and potentially confusing. Wait, looking closer: f(x) = e^x - x - 1. e^x = 1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120 + ... So f(x) = (1 + x + x^2/2 + ...) - x - 1 = x^2/2 + x^3/6 + x^4/24 + x^5/120 + ... The first four nonzero terms are indeed x^2/2, x^3/6, x^4/24, x^5/120. The stated answer lists them as x^5/120 + x^4/24 + x^3/6 + x^2/2. This is the same set. Is it 'wrong'? It's unconventional. But let's look at the solution steps. Step 1 is a sentence. Step 2 is an equation. The equation checks that the difference between f(x) and the polynomial P(x) = -x^5/120 - ... - x^2/2 - x + e^x - 1 goes to 0 faster than x^5? No, the equation is Limit((f(x) - P(x))/x^5) = 0? Let's check the equation content: `Limit((-x**5/120 - x**4/24 - x**3/6 - x**2/2 - x + exp(x) - 1)/x**5, x, 0, dir='+') = 0`. The numerator is `exp(x) - 1 - x - (x^2/2 + x^3/6 + x^4/24 + x^5/120)`. This is the remainder term R_5(x). The limit of R_5(x)/x^5 is 0? No, the limit of R_n(x)/x^n is 0 for Taylor series. Here n=5. The term x^5/120 is included in the subtraction. So the numerator is O(x^6). The limit of O(x^6)/x^5 is 0. So the equation is correct. The stated answer is just the terms in reverse order. This is 'misleading' or 'style' at worst. But wait, the prompt asks for 'first four nonzero terms'. Usually, this implies ascending order. Listing them descending is weird but not mathematically 'wrong' in terms of value. However, the solution sentence 'Start from a known series ... and substitute or multiply' is very vague. It doesn't actually show the work. But the instructions say 'check that ... each conclusion follows from the lines before it'. The conclusion (the stated answer) is not derived in the text, only verified by the equation. The equation verifies that the polynomial matches f(x) up to order 5. It does not verify that these are the *first four nonzero* terms (i.e., that lower order terms are zero). For instance, if the answer included a constant term, the limit check might still pass if the constant was wrong? No, if the constant was wrong, the limit as x->0 of (f(x)-P(x))/x^5 would be undefined or infinite. So the limit check implicitly verifies the lower terms too. The main issue is the ordering. I will mark it as 'misleading' because standard convention is ascending powers.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-10-04 — [domain objection, downgraded to style] The stated answer lists terms in descending order of power (x^5, x^4, ...), which is not the standard convention for Maclaurin series (ascending order). More critically, the stated answer includes x^2, x^3, x^4, and x^5, but the function f(x) = e^x - x - 1 has a nonzero x^2 term (1/2 x^2). The 'first four nonzero terms' should be 1/2 x^2 + 1/6 x^3 + 1/24 x^4 + 1/120 x^5. The provided answer is just a permutation of these, but typically 'first four' implies starting from the lowest degree. However, the main issue is that the solution text is vague ('substitute or multiply') and the equation check is for the limit of the remainder, which is a valid check but doesn't explain the derivation. The biggest error is that the stated answer is technically correct as a set of terms, but the ordering is non-standard and potentially confusing. Wait, looking closer: f(x) = e^x - x - 1. e^x = 1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120 + ... So f(x) = (1 + x + x^2/2 + ...) - x - 1 = x^2/2 + x^3/6 + x^4/24 + x^5/120 + ... The first four nonzero terms are indeed x^2/2, x^3/6, x^4/24, x^5/120. The stated answer lists them as x^5/120 + x^4/24 + x^3/6 + x^2/2. This is the same set. Is it 'wrong'? It's unconventional. But let's look at the solution steps. Step 1 is a sentence. Step 2 is an equation. The equation checks that the difference between f(x) and the polynomial P(x) = -x^5/120 - ... - x^2/2 - x + e^x - 1 goes to 0 faster than x^5? No, the equation is Limit((f(x) - P(x))/x^5) = 0? Let's check the equation content: `Limit((-x**5/120 - x**4/24 - x**3/6 - x**2/2 - x + exp(x) - 1)/x**5, x, 0, dir='+') = 0`. The numerator is `exp(x) - 1 - x - (x^2/2 + x^3/6 + x^4/24 + x^5/120)`. This is the remainder term R_5(x). The limit of R_5(x)/x^5 is 0? No, the limit of R_n(x)/x^n is 0 for Taylor series. Here n=5. The term x^5/120 is included in the subtraction. So the numerator is O(x^6). The limit of O(x^6)/x^5 is 0. So the equation is correct. The stated answer is just the terms in reverse order. This is 'misleading' or 'style' at worst. But wait, the prompt asks for 'first four nonzero terms'. Usually, this implies ascending order. Listing them descending is weird but not mathematically 'wrong' in terms of value. However, the solution sentence 'Start from a known series ... and substitute or multiply' is very vague. It doesn't actually show the work. But the instructions say 'check that ... each conclusion follows from the lines before it'. The conclusion (the stated answer) is not derived in the text, only verified by the equation. The equation verifies that the polynomial matches f(x) up to order 5. It does not verify that these are the *first four nonzero* terms (i.e., that lower order terms are zero). For instance, if the answer included a constant term, the limit check might still pass if the constant was wrong? No, if the constant was wrong, the limit as x->0 of (f(x)-P(x))/x^5 would be undefined or infinite. So the limit check implicitly verifies the lower terms too. The main issue is the ordering. I will mark it as 'misleading' because standard convention is ascending powers.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (style) 2026-10-04 — [domain objection, downgraded to style] The stated answer lists terms in descending order of power (x^5, x^4, x^3, x^2), which is not the standard convention for Maclaurin series (ascending order). More critically, the stated answer is missing the x^1 term. The expansion of e^x - 1 - x is x^2/2 + x^3/6 + x^4/24 + x^5/120 + ... The term -x in f(x) cancels the x term from e^x, so the first nonzero term is x^2/2. The stated answer includes x^2/2 but lists it last, and implies these are the 'first four' by including x^5, but omits the correct ordering and potentially confuses the count. Actually, looking closely: f(x) = -x + (1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120 + ...) - 1 = x^2/2 + x^3/6 + x^4/24 + x^5/120 + ... The first four nonzero terms are x^2/2, x^3/6, x^4/24, x^5/120. The stated answer lists exactly these four terms, but in reverse order. While mathematically the set of terms is correct, presenting a power series in descending order is highly non-standard and misleading for a 'Maclaurin series' problem which implies ascending powers. Furthermore, the solution text is vague ('substitute or multiply') and doesn't explicitly show the cancellation of the linear term, which is the key step. However, the biggest issue is the ordering. A Maclaurin series is defined as sum a_n x^n. Writing it as x^5/120 + ... is not the standard form. Is it an 'error'? It's technically a representation of the same function, but in the context of 'find the first four nonzero terms', the order matters for identification. Usually, 'first' implies lowest degree. Listing highest degree first is confusing. Let's look at the 'Stated answer' again. It has x^5, x^4, x^3, x^2. These are the first four nonzero terms. The order is wrong. Is it an error? In many contexts, yes, because 'first' refers to the index n. n=2,3,4,5. Listing n=5 first is incorrect usage of 'first'. Also, the solution step 1 is just a sentence, not a derivation. The equation checks the limit, which confirms the terms are correct up to x^5. But the presentation of the answer is flawed. I will mark it as misleading/error because the order of terms in a Taylor/Maclaurin series is conventionally ascending, and 'first' refers to the lowest powers.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.