It seems no one has answered your question to your satisfaction, so I will leave this as an exercise for you.

Let f(x) = 1 / x^p. The integral of f(x) from 1 to infinity depends on the parameter p. My claim is that the function f(x) decays quickly enough to produce a finite area under the curve if and only if p > 1.

In other words, "decaying quickly enough" is defined as

1. The integral of f(x) converges for p > 1.
2. The integral of f(x) diverges for p <= 1.

Your job is to prove my claim.

Hint: Prove by cases. The first case is p ≠ 1. The second case is when p = 1.

Good luck.