Answer
$$Convergent,\frac{1}{36}$$
Work Step by Step
$$\int_{1}^{\infty }\frac{1}{(2x+1)^{3}}dx=\lim_{t\rightarrow \infty}\int_{1}^{t}\frac{1}{(2x+1)^{3}}dx$$
$$=\lim_{t\rightarrow \infty} \left |-\frac{1}{4(2x+1)^{2}}\right |_{1}^{t}$$
$$= \lim_{t\rightarrow \infty} (-\frac{1}{4(2t+1)^{2}}+\frac{1}{36})$$
$$=\frac{1}{36}$$
Convergent