Does $\int_{3}^{\infty}\frac{1}{(x-2)^{3/2}}\text{ d}x$ converge? I am trying to see whether or not 
$$\int\limits_{3}^{\infty}\dfrac{1}{(x-2)^{3/2}}\text{ d}x$$
converges. My first instinct was to notice in $[3, \infty)$ that 
$$\dfrac{1}{(x-2)^{3/2}} > \dfrac{1}{x^{3/2}}\text{.} $$
But $\displaystyle\int\limits_{3}^{\infty}\dfrac{1}{x^{3/2}}\text{ d}x$ converges, so that does not give me any helpful information. 
As I started typing this question, I thought of the following idea: does it suffice to show that
$$\lim\limits_{t\to \infty}\int\limits_{3}^{t}\dfrac{1}{(x-2)^{3/2}}\text{ d}x$$
exists?

$$\int\limits_3^\infty\frac1{(x-2)^{3/2}}dx=\left.-2(x-2)^{-1/2}\right|_3^\infty:=-2\lim_{b\to\infty}\left(\frac1{\sqrt{b-2}}-1\right)=2$$
