$\"\"$

Alternating Series Test :

If the alternating series $\"\"$ satisfies

(i) $\"\"$ ,

(ii) $\"\"$ , then the series is convergent.

$\"\"$

The series is $\"\"$ .

Since $\"\"$ .

Rewrite the series as $\"\"$ .

The function $\"\"$ is decreasing because $\"\"$ is an increasing function.

Therefore, the sequence $\"\"$ is eventually decreasing.

Thus, condition (i) is verified.

$\"\"$

Find $\"\"$ .

$\"\"$

Apply L hospital rule.

$\"\"$

$\"\"$ .

Thus, condition (ii) is verified.

Therefore, the given series is convergent by the Alternating Series Test.

$\"\"$

The series $\"\"$ is convergent.