Question

Sum of series $1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+...$ to $\infty .$

Answer 1

Let $S=1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+..\left(1\right)$
$\therefore \frac{S}{5}=\frac{1}{5}+\frac{4}{5^2}+\frac{7}{5^3}+\frac{10}{5^4}+..\left(2\right)$
$\left(1\right)-\left(2\right)$
$\frac{4S}{5}=1+\frac{3}{5}(1+\frac{1}{5}+\frac{1}{5^2}+...............)=1+\frac{3}{5}\frac{1}{1-1/5}1+\frac{3}{4}=\frac{7}{4}$
$\therefore S=\frac{35}{16}$