There are many fractions with a numerator of 1 whose decimal expansions don’t go on to infinitely many decimal places. For example, 1/4 is equivalent to the decimal 0.25, and 1/500 is equivalent to 0.002. However, the decimal expansion of 1/3 is 0.33333 …, a decimal that never terminates.
If you were to add up all these numbers — fractions with a numerator of 1 whose decimal expansions don’t go on forever — what would be the sum? (Note: Before you ask, let’s include the fraction 1/1 in this group.)
Every terminating fraction can be expressed as $\dfrac{m}{10^n}$.
In order to be simplified to have a numerator of 1, the denominator must be in the form of $2^p \cdot 5^q$.
The sum of all these fractions can be represented as,
$$\left(\dfrac{1}{2^0} + \dfrac{1}{2^1} + \dfrac{1}{2^2} + \dots\right) \cdot \left(\dfrac{1}{5^0} + \dfrac{1}{5^1} + \dfrac{1}{5^2} + \dots\right)$$
$$ = \sum\limits_{p = 0}^{\infty} \dfrac{1}{2^p} \cdot \sum\limits_{q = 0}^{\infty} \dfrac{1}{5^q}$$
$$ = \dfrac{1}{1 - \dfrac{1}{2}} \cdot \dfrac{1}{1 - \dfrac{1}{5}}$$