The function is .

.

The power series form of .

.

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 .

Graph :

Graph the function .

The power series of is .

Graph the partial sums , , and .

Observe the graph :

As the value of increases the approximation gets closer to over a wider interval.

The power series of is and radius of convergence is . \ \

Graph of the function .

Graph the partial sums are , , and .

As the value of increases the approximation gets closer to over a wider interval.