p-test for convergence

The p-test is used to determine the convergence or divergence of improper integrals of the form $\int \frac{1}{x^p}dx$. It is the fundamental "borderline" test for integrals involving power laws.

1. The Two Types of Improper Integrals

There are two distinct cases because the behavior of $\frac{1}{x^p}$ changes drastically depending on whether we are looking at infinity (tail behavior) or at zero (singularity behavior).

Type 1: Infinite Intervals (At Infinity)

Integral: ${\int }_1^{\mathrm{\infty }}\frac{1}{x^p}dx$

• Focus: What happens as $x\to \mathrm{\infty }$?
• Converges if: $p>1$
• Diverges if: $p\le 1$

Type 2: Singularities (Near Zero)

Integral: ${\int }_0^1\frac{1}{x^p}dx$

• Focus: What happens as $x\to 0^{+}$? (Vertical asymptote)
• Converges if: $p<1$
• Diverges if: $p\ge 1$

The Harmonic "Wall"

The harmonic function $\frac{1}{x}$ (where $p=1$) is the critical boundary. It fails at both ends: it diverges at infinity (too slow to decay) and diverges at zero (too fast to be tamed).