Answer
Convergent $\;,\; 2$
Work Step by Step
Let \[I=\int_{3}^{\infty}\frac{1}{(x-2)^{\frac{3}{2}}}dx\]
\[I=\lim_{t\rightarrow \infty}\int_{3}^{t}\frac{1}{(x-2)^{\frac{3}{2}}}dx \;\;\; \ldots (1)\]
\[I=\lim_{t\rightarrow \infty}\left[-2(x-1)^{\frac{-1}{2}}\right]_{3}^{t}\]
\[I=\lim_{t\rightarrow \infty}\left[\frac{-2}{\sqrt{t-2}}+\frac{2}{\sqrt{3-2}}\right]\]
\[I=0+2=2\]
Since limit on R.H.S. of (1) is finite so $I=2$
Hence \[I=\int_{3}^{\infty}\frac{1}{(x-2)^{\frac{3}{2}}}dx=2.\]
