The function is .

Consider .

Apply the power series formula: .

Apply derivative on each side with respect to .

Again apply derivative on each side.

Multiply on each side.

Substitute .

Since the first term() of is zero, .

.

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 .