Step 1:

The function is .

Definition of Taylor series:  

If a function has derivatives of all orders at then the series  

is called Taylor series for at .

First find the successive derivatives of .

The n^th derivative of the function is .

Step 2:

The series is centered at .  

Step 3:

Taylor series centered at .

.

Step 4:

Radius of convergence :

By the Ratio Test, the series converges if .

n^th term of the taylor series is .

(n+1)^th term of the taylor series is .

Condition for convergence : .

So the region of convergence is .

Therefore the radius of convergence is R = 2.

Solution :

Taylor series of is .

The radius of convergence is R = 2.