- #1
s3a
- 818
- 8
Homework Statement
The Taylor series for f(x) = ln(sec(x)) at a = 0 is ##Σ_{n=0}^{∞} c_n x^n##.
(a) Find the first few coefficients. (I don't need help for this part.)
(b) Find the exact error in approximating ln(sec(-0.1)) by its fourth-degree Taylor polynomial at a = 0.
Homework Equations
E(x) = f(x) - ##P_n (x)##, or in this case, E(-0.1) = f(-0.1) - ##P_4 (-0.1)##
The Attempt at a Solution
I found the cofficients of each term of the polynomial, but I don't feel it's necessary to show you how I did that, since you can just confirm that I'm right by looking here.:
http://www.wolframalpha.com/input/?i=ln(sec(x))+maclaurin+polynomial
If I'm correct, for this problem, the error is the difference between the function and the 4th degree polynomial.:
E(-0.1) = f(-0.1) - ##P_4 (-0.1)## = ln(sec(-0.1)) – (1/2*0.1^2+1/12*0.1^4) = 2.22899019757457996644E-8 ≠ 2.22899018157481E-8
Notice how my answer (which is the one on the left-hand side) differs slightly from the correct answer (which is the one on the right-hand side) of the problem.
Where is this small disagreement in error/remainder values coming from?
Any input would be greatly appreciated!