136.275, Assignment No. 1

September 18, 2002

The assignment is due Friday, September 25, 2002 in class. Late assignments receive a mark zero.

1. a) Prove by using the definition of the limit that if 0 r <1, we have that

. [6]

b) State the theorem by which part a) implies that for –1<r 0 also. [2]

2. Determine if the sequence converges or not, and if it does, find the limit:

a) , [4]

b) , [4]

c) . [5]

3. a) Prove that a sequence converges to a limit L if and only if the sequence of

even terms and the sequence of odd terms both converge to L. [8]

b) For which values of a does the sequence a, 0 , , 0, , 0 , … converge?

Explain. [2]

4. a) Give an example of a sequence to show that the convergence of { |a_n| } does

not imply the convergence of { a_n } . [2]

b) Use the Squeeze Theorem to show the convergence of the sequence . [4]

c) Consider the sequence {a_n} whose n-th term is . Show that

by interpreting as the Riemann sum of a definite integral. State

all of the theorems on integrals that you are using. [5]

Total [ 42]