Why does this converge or diverge? (Improper Integral)

Question

I am trying to find the interval of convergence for this integral. I am falling flat on the last step, and do not understand why the value of p matters (Except that p=1 causes a div by zero)

Answer 1

Hint:

$$\lim_{b\to 0^+} b^e=\infty$$

For $e<0$ and

$$\lim_{b\to 0^+} b^e=0$$

For $e>0$

Also for $p=1$, you have to recall the antiderivative of $\int \frac{1}{x} \ dx$