# Taylor Polynomial

1. Jun 15, 2009

### wimma

1. The problem statement, all variables and given/known data
(a) Give Taylor Polynomal of order 4 for ln(1+x) about 0.
(b) Write down Tn(x) of order n by looking at patterns in derivatives in part (a), where n is a positive integer.
(c) Write down the remainder term for the poly. in (b)
(d) How large must n be to ensure Tn gives a value of ln(1.3) which has an error less than 0.0002

Question 2 part d

2. Relevant equations

3. The attempt at a solution
Okay so we have that the taylor poly is Tn(x) = ∑[f'(a)(x-a)^n]/n! where f' is the nth derivative
And the remainder is Rn(x) = f'(c)(x-a)^(n+1)/(n+1)! where f' is the n+1 th derivative and c lies between x and a
For part a I differentiated ln(1+x) a few times and got a pattern... the values I got at x=0 were 0, 1, -1, 2, -6, 25, -120.
and I got T4 = x - x^2 /2 + x^3 /3 -x^4 /4
b. I got the highest term as (-1)^(n+1) x^n /n
c. I got the remainder term to be (-1)^n (x^(n+1)/(1+c)^(n+1)*(n+1))
d. I got an answer of n=4999 using the approximation that abs(Rn) is always less than abs(1/(n+1))

Last edited: Jun 15, 2009
2. Jun 15, 2009

### Cyosis

I am afraid I don't have a log in for the university of Sydney.

3. Jun 15, 2009

### wimma

4. Jun 15, 2009

### Cyosis

You want to be taken through the entire exercise, which is fine. However it is not fine to request that without showing your work. So lets start with that, show your work for a b c, write down the formulae you think are relevant. Like Taylor's theorem the definition of the remainder etc.

5. Jun 15, 2009

Updated