> [!note] Definition > > $ > \int_{a}^{\infty}\frac{1}{x^p}dx > $ > The p-integral is an [[Improper Integral|improper integral]] of a function $f(x) = \frac{1}{x^p}$ from some value $a$ to $\infty$. The integral converges to a finite value if $p > 1$, and diverges to infinity if $p \le 1$. > $ > \begin{align*} > \text{if}\ p \gt 1 \text{ then } &\int_{a}^{\infty}\frac{1}{x^p}dx \\ > &= -\frac{1}{(p - 1)x^{p - 1}}\bigg|^{\infty}_{a} \\ > &= \frac{1}{(p - 1)a^{p - 1}} \\ > \text{if}\ p \le 1 \text{ then } &\int_{a}^{\infty}\frac{1}{x^p}dx = \infty \\ > \end{align*} > $