$\"\"$

\

Ratio Test :

\

(i) If $\"\"$ , then the series is $\"\"$ is absolutely convergent.

\

(ii) If $\"\"$ or $\"\"$ , then the series is $\"\"$ is divergent.

\

(iii) If $\"\"$ , then the ratio test is inconclusive. $\"\"$

\

The integral is $\"\"$ .

\

Rewrite the integral.

\

$\"\"$ .

\

The power series form of $\"\"$ is $\"\"$ .

\

$\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

Theerefore, the power series is $\"\"$ . $\"\"$

\

The series is $\"\"$

\

Consider $\"\"$ .

\

$\"\"$

\

$\"\"$

\

Substitute $\"\"$ in $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

If $\"\"$ then $\"\"$ .

\

$\"\"$

\

$\"\"$

\

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

\

$\"\"$ .

\

The radius of convergence is $\"\"$ .

\

The series is convergence in the interval $\"\"$ .

\

Therefore, the power series is $\"\"$ and $\"\"$ . $\"\"$

\

The power series is $\"\"$ and $\"\"$ .