$\"\"$

\

The function is $\"\"$ .

\

The function $\"\"$ is the derivative of $\"\"$ .

\

Power series formula: $\"\"$ .

\

$\"\"$

\

$\"\"$

\

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

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

The power series representation 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 $\"\"$ .

\

Consider $\"\"$ and $\"\"$ .

\

Find $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

The series is converges $\"\"$ .

\

$\"\"$ .

\

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

\

$\"\"$

\

(b)

\

Find the power series for $\"\"$ .

\

The function is $\"\"$ .

\

The function $\"\"$ is the derivative of $\"\"$ .

\

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

\

$\"\"$

\

$\"\"$

\

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

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

The first term of the series is $\"\"$ .

\

$\"\"$ .

\

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

\

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

\

The derivative of $\"\"$ function also have same radius of convergence.

\

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

\

$\"\"$

\

(c)

\

Find the power series for $\"\"$ .

\

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

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

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

\

Find the radius of convergence.

\

Consider $\"\"$ .

\

Apply ratio test:

\

Consider $\"\"$ and $\"\"$ .

\

Find $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

The series is converges $\"\"$ .

\

$\"\"$ .

\

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

\

$\"\"$

\

(a) The power series representation of $\"\"$ is $\"\"$ and $\"\"$ .

\

(b) The power series of $\"\"$ is $\"\"$ and $\"\"$ .

\

(c) The power series of $\"\"$ is $\"\"$ and $\"\"$ .