Instructions: Each question will be graded both for correctness and for work shown. The latter is extremely important solutions should be written using proper English (that means, appropriate grammar and punctuation!) and should include a full, clear explana tion, Do not use a calculator (physical or online) unless explicitly asked to. Context: For much of this written assignment, we will explore various uses of Taylor polynomials. You may take for granted the following: Given a function f : R - R and its Taylor polynomial Pr expanded around a = 0, it is a fact that there is c in between' 0 and z such that (n + 1)! (E) Here f(+1) is the (n + 1)th derivative of f, and recall that PR(c) = M: k! 2 n! You may recognize the n = 0 case as the "Mean Value Theorem." Problem 1. (4) Using (E) with n = 1 and f(x) = In(1 + 2), show that In(1 ty) Q. (ii) Using ( E) with n = 2 and f(c) = In(1 + 2), show that 3 - 2 Q.(iii) Use (i) and (it) to deduce that, for every z ( [0, 1], 2 2 - I 1-1/2n. Conclude that " * " ) " - ( 1 + 4 ) " > et-4am. Observed that n+1 n lim = E. (ini) Show that the series n-1 converges. Deduce that lim n! 1400 mx+1 . =0. (In other words, while n! grows faster than a" for any I, it does not grow faster than (iv) Show that the series 3* n! converges. You may find it helpful to use (ii). Deduce that lim 2 400 327! = Q. (In other words, while n does not grow as fast as not, it does grow faster than (n/3)")