Question

For $|x|<1$, the constant term in the expansion of $\frac{1}{(x-1)^{2}(x-2)}$ is

• A. 2
• B. 1
• C. 0
• D. $-\frac{1}{2}$

Answer 1

The Correct Option is D

$\frac{1}{(x-1)^{2}(x-2)} =\frac{1}{-2(1-x)^{2}\left(1-\frac{x}{2}\right)}$ $=-\frac{1}{2}\left[(1-x)^{-2}\left(1-\frac{x}{2}\right)^{-1}\right]$ $=-\frac{1}{2}\left[(1+2 x+\ldots)\left(1+\frac{x}{2}+\ldots\right)\right]$ $\therefore$ Coefficient of constant term is $-\frac{1}{2}$