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 series is centered at .

Taylor series centered at .

.

The derivative of the function is .

Radius of convergence :

By the Ratio Test, the series converges if .

term of the taylor series is .

term of the taylor series is .

If then .

Condition for convergence : .

So the region of convergence is .

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

.

Therefore the radius of convergence is .

Taylor series of  is .

The radius of convergence is .