Question

Find the Taylor series for f (x) centered at the given value of a . [Assume that f has a power series expansion. Do not show that Rn (x) ----->0
.] Also find the associated radius of convergence.

f (x) = ln x, a = 2

Answer 1

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 .

Answer 2

Contd....

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.