Question

Find the coefficient of $x^r$ in the expansion of $\frac{1-ax-x^2}{e^x}$.

Answer 1

Consider,

$\frac{1-ax-x^2}{e^x}$

$=(1-ax-x^2)e^{-x}$
$=(1-ax-x^2)\{1-x+\frac{x^2}{|\underset{–}{2}}-\frac{x^3}{|\underset{–}{3}}+...+\frac{(-1{)}^r{x}^r}{|\underset{–}{r}}+...\}$.
Therefore, the required coefficient $=\frac{(-1{)}^r}{|\underset{–}{r}}-\frac{(-1{)}^{r-1}a}{|\underset{–}{r-1}}-\frac{(-1{)}^{r-2}}{|\underset{–}{r-2}}$
$=\frac{(-1{)}^r}{|\underset{–}{r}}\{1+ar-r(r-1\left)\right\}$