Question

Find the sum of series
$0.5+0.55+0.555+...$

Answer 1

$S=0.5+0.55+0.55+....$

$⟹S=5(0.1+0.01+0.001+...)$

The series $0.1,0.01,0.01$ is the G.P. with first term $a=0.1$ and common ratio $r=\frac{0.01}{0.1}=0.1$

We know that the sum of infinite series is $\frac{a}{1-r}$

$⟹S=5\left(\frac{0.1}{1-0.1}\right)$

$⟹S=5\left(\frac{0.1}{0.9}\right)$

$⟹S=\frac{5}{9}$