$\"\"$

The series is $\"\"$ .

Consider $\"\"$ .

The $\"\"$ -series $\"\"$ is convergent if $\"\"$ and divergent if $\"\"$ .

The series $\"\"$ is convergent if and only if $\"\"$ .

Since the series $\"\"$ is an alternating series, it is convergent when $\"\"$ .

Therfore, $\"\"$ is convergent for $\"\"$ .

The series is convergent for $\"\"$ .

$\"\"$

The series is convergent for $\"\"$ .