$\"\"$

The repeating decimal is $\"\"$ .

Rewrite the decimal $\"\"$ .

$\"\"$

In the above series

$\"\"$ $\"\"$

The infinite sum of an geometric series is $\"\"$ , with $\"\"$ ,

Where $\"\"$ is first term and r is common ratio.

$\"\"$ , the sum exists.

$\"\"$ (Substitute $\"\"$ )

$\"\"$ (Subtract denominator: $\"\"$ )

$\"\"$ (Divide: $\"\"$ ) $\"\"$

The sum of the infinite geometric series is $\"\"$ .