Question

If $S=\sum _{n=0}^{\infty }\frac{2n+3}{3^n}$ then, the value of $S$ is equal to

Answer 1

$S=\sum _{n=0}^{\infty }\frac{2n+3}{3^n}$
$\Rightarrow S=\frac{3}{3^0}+\frac{5}{3^1}+\frac{7}{3^2}+\frac{9}{3^3}+...\cdots \left(1\right)$
This is an $A.G.P.$
$\Rightarrow \frac{1}{3}S=\frac{3}{3}+\frac{5}{9}+\frac{7}{3^3}+....\cdots \left(2\right)$
So subtracting (1) and (2) we get,
$\Rightarrow \frac{2}{3}S=3+(\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+......)$
$\Rightarrow \frac{2}{3}S=3+2\left(\frac{\frac{1}{3}}{1-\frac{1}{3}}\right)$
$\Rightarrow \frac{2}{3}S=3+2\times \frac{1}{2}\Rightarrow S=6$