Comparison test for improper integrals
Problem 4.1092 · medium
Use a comparison to decide whether \( \displaystyle \int_1^\infty e^{- x^{2}} \, dx \) converges or diverges.
- For x ≥ 1, x² ≥ x, so e^(−x²) ≤ e^(−x). Both are positive.
- The difference, which is ≥ 0 for x ≥ 1: -exp(-x**2) + exp(-x).
- \[ \int\limits_{1}^{\infty} e^{- x}\, dx = e^{-1} \]The comparison integral converges.✓ Proved
- A smaller positive integrand than a convergent one converges; a larger one than a divergent one diverges. So it converges.
Answer \( \text{converges} \)
Lines: 1 proved, 3 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 | Not checked | — | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | Not checked | — | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the integral over each decade [10ᵏ, 10ᵏ⁺¹] for k up to 7: its pieces vanish only if it converges |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "pass", "severity": "", "notes": "The solution correctly applies the Direct Comparison Test. The inequality $e^{-x^2} \le e^{-x}$ for $x \ge 1$ is valid, and the convergence of the compar
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-17 — reviewer response could not be parsed: { "verdict": "pass", "severity": "", "notes": "The solution correctly applies the Direct Comparison Test. The inequality $e^{-x^2} \le e^{-x}$ for $x \ge 1$ is valid, and the convergence of the compargpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: { "verdict": "pass", "severity": "", "notes": "The solution correctly applies the Direct Comparison Test. The inequality $e^{-x^2} \le e^{-x}$ for $x \ge 1$ is valid, and the convergence of the compargpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: {"verdict": "pass", "severity": "", "notes": "The solution correctly applies the Direct Comparison Test. The inequality $e^{-x^2} \le e^{-x}$ for $x \ge 1$ is valid, and the convergence of the comparigpt-oss:20b: pass 2026-10-14
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/improper_comparison, checked 2026-10-17 with SymPy 1.14.0.