Question

The sum of the infinite series: $\frac{1}{10}+\frac{2}{{10}^2}+\frac{3}{{10}^3}+....+\frac{n}{{10}^n}+.....$

• A. $\frac{1}{9}$
• B. $\frac{10}{81}$
• C. $\frac{1}{8}$
• D. $\frac{17}{22}$

Answer 1

Answer: (B)

$\frac{10}{81}$

$S=\frac{1}{10}+\frac{2}{{10}^2}+\frac{3}{{10}^3}+....+\frac{n}{{10}^n}+.....\infty$
$\frac{S}{10}=\frac{1}{{10}^2}+\frac{2}{{10}^3}+.....\infty$
Subtracting,
$\frac{9S}{10}=\frac{1}{10}+\frac{1}{{10}^2}+\frac{1}{{10}^3}+......\infty$
$\frac{9S}{10}=\frac{\frac{1}{10}}{1-\frac{1}{10}}$
$\frac{9S}{10}=\frac{1}{9}$
$S=\frac{10}{81}$