Question

What is the sum of the geometric series 1500 + 150 + 15 + 1.5 + 0.15 + 0.015, ... ? in a fraction

Answer 1

$1500+150+15+\cdots=1500\left(1+\dfrac1{10}+\dfrac1{100}+\cdots\right)$

The geometric sum can be written as

$\displaystyle1500\sum_{n=0}^\infty \frac1{10^n}=\frac{1500}{1-\frac1{10}}=\dfrac{5000}3$