can you please help me with this Taylor series problem???????????

Question

Use zero- through fourth-order Taylor series expansions to predict f(2.5) for: f(x) = ln x using a base point at x = 1. Compute the true percent relative error E% for each approximation. Discuss the meaning of the results

Answer 1

f(x) = ln(x) ----> f(1) = ln(1) = 0

 

f'(x) = 1/x = x^(-1)  ---> f'(1) = 1

 

f''(x) = -x^(-2) = -1/x^2  ---> f''(1) = -1

 

f'''(x) = 2x^(-3) ---> f'''(1) = 2

 

f4(x) = -6x^(-4) ---> f4(1) = -6

 

f5(x) = 24x^(-5)

 

 

P(x) = f(1)/0! *x^0  + f'(1)/1!*x  + f''(1)/2!*x^2 + f'''(1)/3!*x^3 + f4(1)*x^4/4!

    

         = x - (1/2)x^2 + (1/3)x^3 + (1/4)x^4  + ... + (1/n)x^n