$\"\"$

\

The series is $\"\"$ .

\

The series can be written as $\"\"$ .

\

Change the series from decimals to fractions :

\

$\"\"$ .

\

$\"\"$

\

Consider $\"\"$ .

\

For $\"\"$ :

\

$\"\"$

\

$\"\"$

\

This is a geomentric series with initial term $\"\"$ and common ratio $\"\"$ .

\

Therefore the sum is $\"\"$ .

\

$\"\"$

\

$\"\"$ .

\

$\"\"$

\

Consider $\"\"$ .

\

For $\"\"$ :

\

$\"\"$

\

$\"\"$

\

This is a geomentric series with initial term $\"\"$ and common ratio $\"\"$ .

\

Therefore the sum is $\"\"$ .

\

$\"\"$

\

$\"\"$ .

\

Because both the series are convergent subtract the first series from the second one.

\

$\"\"$

\

The series converges and sum of the series is $\"\"$ .

\

$\"\"$

\

The series converges and sum of the series is $\"\"$ .