Question

The coefficient of $x^3$ in the infinite series expansion of $\frac{2}{(1-x)(2-x)}$, for $\left|x\right|<1$ is

• A. $-\frac{1}{16}$
• B. $\frac{15}{8}$
• C. $-\frac{1}{8}$
• D. $\frac{15}{16}$

Answer 1

The correct option is B $\frac{15}{8}$

Given expression can be rewritten as $2{(1-x)}^{-1}\times 2^{-1}{(1-\frac{x}{2})}^{-1}$

$=[1+x+x^2+x^3+\dots ][1+\left(\frac{x}{2}\right)+{\left(\frac{x}{2}\right)}^2+{\left(\frac{x}{2}\right)}^3+\dots ]$$\therefore$ Coefficient of $x^3$ in the above expression is

$=[{\left(\frac{1}{2}\right)}^3+{\left(\frac{1}{2}\right)}^2+\left(\frac{1}{2}\right)+1]$

$=[\frac{1}{8}+\frac{1}{4}+\frac{1}{2}+1]$

$=\left[\frac{1+2+4+8}{8}\right]=\frac{15}{8}$