$\"\"$

\

The function is $\"\"$ .

\

Power series formula: $\"\"$ .

\

Consider $\"\"$ .

\

Replace $\"\"$ by $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

$\"\"$

\

$\"\"$

\

Apply derivative on each side with respect to $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

Again apply derivative on each side with respect to $\"\"$ .

\

$\"\"$

\

$\"\"$

\

Multiply $\"\"$ on each side.

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

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

\

$\"\"$

\

Find the radius of convergence.

\

Consider $\"\"$ .

\

Ratio test :

\

Let $\"\"$ be a series with non zero terms.

\

1. $\"\"$ converges absolutely if $\"\"$ .

\

2. $\"\"$ diverges if $\"\"$ or $\"\"$ .

\

3. The ratio test is inconclusive if $\"\"$ .

\

Here $\"\"$ and $\"\"$ .

\

Find $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

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

\

$\"\"$

\

$\"\"$ .

\

Therefore, the radius of the convergence is $\"\"$ .

\

$\"\"$

\

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

\

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