Find the Taylor polynomials of degree 3 for f(x) = x^3 + 2x^2 + x - 2 about a = 1? Also explain the error in this polynominal

Question

I have already found out the Taylor polynomial of degree 3

 

f(x) = x³ + 2x² + x − 2 ----> f(1) = 2
f'(x) = 3x² + 4x + 1 -------> f'(1) = 8
f''(x) = 6x + 4 --------------> f''(1) = 10
f'''(x) = 6 --------------------> f'''(1) = 6

f(x) = 2/0! + 8/1! (x−1) + 10/2! (x−1)² + 6/3! (x−1)³
f(x) = 2 + 8(x−1) + 5(x−1)² + (x−1)³

 

How can I explain or find the error?  Would the Taylor series based on the same approach converge?

Answer 1

So after three steps, f(x) using the Taylor expansion around x=1 is:

 

2 + 8(x−1) + 5(x−1)² + (x−1)³ =

2 + 8x - 8 + 5(x² - 2x + 1) + x³ - 3x² + 3x - 1 =

8x - 6 + 5x² - 10x + 5 + x³ - 3x² + 3x -1 = 

x³ + 2x² + x - 2.

 

That's the same as the original polynomial.

 

So what is the error you're concerned about?