$\frac{\frac{1}{2}\cdot \frac{2}{2}}{1^3}+\frac{\frac{2}{2}\cdot \frac{3}{2}}{1^3+2^3}+\frac{\frac{3}{2}\cdot \frac{4}{2}}{1^3+2^3+3^3}+\dots ..$ up to n terms is equal to

For n =1, we have

$\frac{\mathrm{n}}{\mathrm{n}+1}=\frac{1}{2}\text{and}\frac{\frac{1}{2}\cdot \frac{2}{2}}{1^3}=\frac{1}{2}$

For n = 2, we have

$\frac{\mathrm{n}}{\mathrm{n}+1}=\frac{2}{3} \text{and }\frac{\frac{1}{2}\cdot \frac{2}{2}}{1^3}+\frac{\frac{2}{2}\cdot \frac{3}{2}}{1^3+2^3}=\frac{1}{2}+\frac{3}{2\times 9}=\frac{2}{3}$

Hence, option (b) is correct.