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Homework Statement
(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
Homework Equations
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))
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