Question

The sum of the infinite series $\frac{1.2}{3!}+\frac{{2.2}^2}{4!}+\frac{{3.2}^3}{5!}+\cdots \cdots$ is

• A. $\frac{2}{3}$
• B. $\frac{3}{2}$
• C. $\frac{3}{4}$
• D. 1

Answer 1

The correct option is D 1

$a_r=\frac{r{.2}^r}{(r+2)!}=\frac{(r+2-2)2^r}{(r+2)!}=\frac{(r+2)2^r}{(r+2)!}-\frac{2^{r+1}}{(r+2)!}=\frac{2^r}{(r+1)!}-\frac{2^{r+1}}{(r+2)!}\therefore {\sum }_{r=1}^n{a}_r=\frac{2}{(1+1)!}-\frac{2^{n+1}}{(n+2)!}=1-\frac{2^{n+1}}{(n+2)!}$
$\therefore$ Sum upto infinite terms = 1