Limits of Riemann sums as integrals
Problem 4.1060 · medium
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{3 + \frac{3 \left(3 i - 3\right)}{n}}{n} \) as a definite integral, and evaluate it.
- Δx = 3/n, and x_i = 1 + 3(i − 1)/n runs across [1, 4] (left endpoints). The sum is Σ f(x_i)Δx with f(x) = x.
- \[ \int\limits_{1}^{4} x\, dx = \frac{15}{2} \]Evaluate the integral.✓ Proved
- \[ \lim_{n \to \infty} \sum_{i=1}^{n} \frac{3 + \frac{9 i - 9}{n}}{n} = \frac{15}{2} \]The limit of the sum itself, from the summation formulas, agrees.✓ Proved
Answer \( \int_{1}^{4} x\, dx = \frac{15}{2} \)
Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the Riemann sum itself with n = 200,000, within its error bound |
Reviewers
gpt-oss:20b: fail (error) — The proposed Riemann sum uses Δx=3/n and f(x)=x, but the given summand simplifies to (9i-6)/n, which does not equal (1+3(i-1)/n)(3/n). Thus the setup does not correctly model the limit as a Riemann sum for ∫₁⁴x dx.qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-17 — reviewer returned a non-objectgpt-oss:20b: fail (error) 2026-10-17 — The proposed Riemann sum uses Δx=3/n and f(x)=x, but the given summand simplifies to (9i-6)/n, which does not equal (1+3(i-1)/n)(3/n). Thus the setup does not correctly model the limit as a Riemann sum for ∫₁⁴x dx.qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer returned a non-objectgpt-oss:20b: fail (error) 2026-10-14 — The first sentence incorrectly identifies Δx and f(x). The given sum does not correspond to a Riemann sum with Δx=3/n and f(x)=x; the algebraic form of the summand does not match the proposed f(x_i)Δx. Thus the setup is wrong.qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer returned a non-objectgpt-oss:20b: inconclusive 2026-10-14 — reviewer returned a non-object
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/riemann_limit_integral, checked 2026-10-17 with SymPy 1.14.0.