The function is  and .

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 .

Apply derivative on each side with respect to .

.

Find the values of the above functions at .

.

The n^th derivative of the function  is .

The series is centered at .

.

Taylor series centered at . 

.

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 .

The radius of convergence is half the width of the interval.

Therefore the radius of convergence is .

Taylor series of  is .

The radius of convergence is .