Referring to part (b):
I don't know what $a$ stands for, and also don't get what "convergence" the author is referring to here. If he means the convergence as $a$ tends to $\infty$, then shouldn't the convergence instead be identical to that in part (a)?
In fact, $a$ is simply intended to be some fixed positive number.
The first two are closely related, as it turns out. Consider the following formal integrals: $$\int_a^\infty x^p\,dx\tag{$\heartsuit$}$$ $$\int_0^b x^p\,dx\tag{$\spadesuit$}$$
It is easy to see that $(\heartsuit)$ diverges when $p\ge 0,$ regardless of which real $a$ is chosen, since $x^p$ will then be (eventually) positive non-decreasing. On the other hand, $(\spadesuit)$ converges when $p\ge 0$ for any real $b,$ as an integral of a bounded continuous function. More interesting is what happens when we are dealing with $p<0.$
Let's suppose that $p<0,$ and that $a,b>0.$ Think about this in terms of the differential "area slices" under the curve.
When $p\ge-1,$ the vertical "area slices" don't get short enough quickly enough as we move to the right for $(\heartsuit)$ to converge. If we were finding the area under the curve with an integral with respect to $dy,$ the horizontal "area slices" grow too long too quickly as we move down. Otherwise, we're fine.
When $p\le-1,$ the vertical "area slices" grow too tall too quickly as we move left toward the $y$-axis for $(\spadesuit)$ to converge. If we were finding the area under the curve with an integral with respect to $dy,$ the horizontal "area slices" don't get narrow enough quickly enough as we move up. Otherwise, we're fine.
Do you see how these behaviors are basically exact mirrors of each other?