MAT 320 Introduction to Analysis

December 19, 1996 Final Examination

B-track

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?

Tony Phillips
March 26 1997