**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?

March 26 1997