Question

I just want to clarify what I need help on (as I do not need help on the full question). Specifically what I want is to be able to set the equations given to me so that I can solve for n using a root finding algorithm (which I already have).

As I understand, I need something of the form | Tn(x) - e^x | = epsilon machine = (x^(n+1) / (n+1)!) and all I need from there is to rearrange the equation so that I can set it equal to zero, plug in a given value of x, and use my root finding algorithm to find the solution (n).

Essentially, I am having trouble deriving the non-linear function of a single variable.

Question 3 - Solve some "real" rootfinding problems Each of the following math problems will lead to a non-linear function of a single variable that will require solution by rootfinding. For each problem, derive the non-linear function and use the rootfinding routine of your choice (or try several), to determine the numerical solution. (a) [8] provide a numerical solution for the number of terms n in the Taylor polynomial expansion of exp(x) (i.e. Tn(x)) such that the relative error is 7+1 r= r =- 1 Emach (n + 1)! Give solutions for x = 10,100, 709, 1000 and compare your numerical solution to the approximate solution given in last homework n > e* x by calculating the relative error between the your numerical solution and your simpler estimate. Note: To make this problem easier, just use Stirling's approximation log(n!) nlog n - n. which will work quite well. However, for a little bit of extra credit solve this for the full problem and compare solutions.

Answer 1

Taylon Theorem it If fis a function continuous and n time, differentiable in an internal im, nich], then there exist some interal int point in this internal denoted by noah for some de roll such that f (outh) > f(a) + hf' the t'em. - - .. i Ý -- + Anni (M-12 tot Tn-1)! no If f is analytic. then derivative of all onden exist and hence. lim hafa) = 0. 1. [Convergence ? fa) about the point n:o gives maclaurian Series f@v »f6% +uffon + "074..-t fo:mm fag=en f(a)= eh tn fo) = 1 . f = 1 If follows that for lalea, we have Taylore Series,