MAT 320 Introduction to Analysis
December 19, 1996 Final Examination
THIS EXAMINATION IS ``OPEN BOOK.'' TOTAL SCORE = 100
1. (20 points) Let
a. Prove by induction that .
b. Prove that
c. Now consider the sequence defined by
Prove that this sequence has a limit L, where .
2. (10 points) Prove that the function is continuous, but not uniformly continuous, on the open interval (0,1).
3. (10 points) Consider the function
Prove carefully and directly that f is Riemann integrable on [0,3]. ``Directly'' means show that f satisfies the definition of Riemann integrability; DO NOT use Theorem 3.3.1!
4. (20 points) Consider the problem of calculating by applying Newton's method to find a root of .
a. Starting with the initial guess work two iterates of the method to calculate and .
b. Give an example of an initial guess that will NOT lead to a sequence of numbers converging to the root.
5. (20 points) Consider the function
a. Show that this function is differentiable at x=0.
b. Show that f is continuously differentiable, i.e. that f' is a continuous function.
6. (20 points) Consider the function .
a. Calculate the first three terms of the Taylor series for f about 0.
b. What is the n-th term in that series?