Question

Find the sum of the following series $\frac{p^r}{\lfloor r}+\frac{p^{r-1}}{\lfloor r-1}\cdot \frac{q}{1}+\frac{p^{r-2}}{\lfloor r-2}\cdot \frac{q^2}{\lfloor 2}+\frac{p^{r-3}}{\lfloor r-3}\cdot \frac{q^3}{\lfloor 3}+.....$ to $r+1$ terms.

Answer 1

$\Rightarrow e^{px}=1+px+\frac{p^2{x}^2}{2!}+.....+\frac{p^{r-1}\cdot x^{r-1}}{(r-1)!}+\frac{p^r{x}^r}{r!}+....$

$e^{qx}=1+qx+\frac{q^2{x}^2}{2!}+.....+\frac{q^{r-1}\cdot x^{r-1}}{(r-1)!}+\frac{q^r{x}^r}{r!}+....$

$S=$co efficient of $x^r$ in $e^{px}\cdot e^{qx}$

$=$coefficient of $x^r$ in $e^{(p+q)x}$

$=\frac{{(p+q)}^r}{r!}$