Comparison test for improper integrals
Problem 4.1093 · medium
Use a comparison to decide whether \( \displaystyle \int_1^\infty \frac{1}{\sqrt{x} - \frac{1}{2}} \, dx \) converges or diverges.
- √x − ½ < √x, so the integrand is larger than 1/√x.
- \[ \frac{1}{\sqrt{x} - \frac{1}{2}} - \frac{1}{\sqrt{x}} = \frac{1}{- \sqrt{x} + 2 x} \]The difference, which is ≥ 0 for x ≥ 1.✓ Proved
- \[ \int\limits_{1}^{T} \frac{1}{\sqrt{x}}\, dx = 2 \sqrt{T} - 2 \]Up to T the comparison integral is this, which grows without bound as T → ∞: it diverges.✓ Proved
- A smaller positive integrand than a convergent one converges; a larger one than a divergent one diverges. So it diverges.
Answer \( \text{diverges} \)
Lines: 2 proved, 2 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The algebraic expression for the difference in step 2 is incorrect; it should be \(\frac{1}{2\sqrt{x}(\sqrt{x}-\tfrac12)}\), not \(\frac{1}{-\sqrt{x}+2x}\qwen3.6:27b-mlx: pass — The solution correctly applies the Direct Comparison Test. The inequality 1/(sqrt(x) - 1/2) > 1/sqrt(x) holds for x > 1/4, and since the comparison integral diverges, the original integral diverges.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the Direct Comparison Test. The inequality 1/(sqrt(x) - 1/2) > 1/sqrt(x) holds for x > 1/4, and since the comparison integral diverges, the original integral diverges.gpt-oss:20b: inconclusive 2026-10-17 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The algebraic expression for the difference in step 2 is incorrect; it should be \(\frac{1}{2\sqrt{x}(\sqrt{x}-\tfrac12)}\), not \(\frac{1}{-\sqrt{x}+2x}\qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the Direct Comparison Test. The inequality 1/(sqrt(x) - 1/2) > 1/sqrt(x) holds for x >= 1, and since the integral of 1/sqrt(x) diverges, the original integral must also diverge. The logic is sound.gpt-oss:20b: fail (misleading) 2026-10-14 — The expression for the difference of the integrands is incorrect, though the conclusion that the integrand is larger than 1/√x is correct. The incorrect algebra could mislead a student about how to compute such differences.qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the Direct Comparison Test. The inequality 1/(sqrt(x) - 1/2) > 1/sqrt(x) holds for x >= 1, and since the integral of 1/sqrt(x) diverges, the original integral must also diverge.gpt-oss:20b: fail (misleading) 2026-10-14 — Step 2 contains an incorrect algebraic simplification of the difference of the integrands; although the conclusion is correct, the erroneous expression could mislead a student into believing the calculation is valid.
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.