136.275, Assignment No. 1

September 19, 2003

The assignment is due Friday, September 26, 2003 in class. Late assignments receive a mark zero.

1. Let {a_n} be a sequence such that = L and let L 0.

a) Prove that eventually a_n 0, i.e. that there exists an N in |N such that for every n N we have that a_n 0. (Hint: Show that, eventually, |a_n| >.) [6]

b) Prove by using the definition of the limit that . [6]

2. Show which of the following sequences is bounded or unbounded by using the definitions:

a) = , [4]

b) = . [4]

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

a) , [4]

b) , [4]

c) . [5]

4. a) Use the Squeeze Theorem to show the convergence of the

sequence . [4]

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

by interpreting as a Riemann sum of a definite integral. Which

theorems on integrals you have to use? [5]

Total [ 42/40]