∫₁^∞ (1/x²) dx converges to 1 while ∫₁^∞ (1/x) dx diverges, although both integrands shrink to zero. Why does going to zero not settle the question?

D. Because what matters is how fast the integrand shrinks.

Evaluate both to b: 1 − 1/b approaches 1, and ln b grows without bound, slowly and without stopping. Shrinking to zero is necessary and not sufficient, and the boundary is sharp, ∫x^(−p) converges exactly when p > 1. It is the harmonic series again, connected directly by the integral test, and it is why Gabriel's horn has infinite area and finite volume.