Math H161 Brown Au03

Homework 1,  due October 1, 2003

 

1.  Formulate your own definitions, analogous to our definitions of upper bound and least upper bound, of

 

            (a)  lower bound of a nonempty set  S

 

(b)  greatest lower bound of a nonempty set   S

 

Hint for (a): try to come up with a suitable definition for sets on your own before you look in the index of our textbook to find its definition for sequences.

 

 

2.  Section 13.2 of our textbook discusses sequences of real numbers.  A sequence is simply a list of real numbers consisting of a first number  x1, a second number  x2, a third x3, and so on, so that there is a real number associated with every positive integer 1, 2, 3,    The positive integer serves as the subscript when the real numbers are written using the notation x1,  x2,  x3,      Real numbers may be repeated at will in a sequence, and you still have a sequence.  On the other hand, you can make the real numbers in a sequence into a set of real numbers by simply ignoring duplications and the order in which the numbers appear in the sequence.

 

For example, the sequence 1, 1, 1, gives you the set  { 1 }, and the sequence 1, 0, 1, 0,  0,   gives you the set  { 0. 1 }.  But the sequence xn, = 1/n,  i.e.,  1, 1/2, 1/3, 1/4,   gives you the set  {1, 1/2, 1/3, 1/4, ]  because there are no duplications at all.

 

(a)  Write the set corresponding to the sequences (d) and (e) in Example 1 on page 432.

 

(b)  Use the formula for the sum of a geometric series to express each xn,  in sequence (f) in a shorter form.

 

(c) Continuing from part (b) above,  write the set corresponding to sequence (f).

 

3. For the sets you found in 2. above, for each of the sets for sequences (d),(e), (f), find the least upper bound of the set (if there is one) and the greatest lower bound of the set (if there is one).

 

OVER for problem 4.


 

 

4.  Let  f(x) = x3. 

(a)  Sketch a graph of this function from x = 0  to x = 1,

 

(b)  Select three points  x1,  x2,  x3,  which will break up the closed interval [0, 1]  into four subintervals for which the length of the longest subinterval is less that 0.3.

 

(c)  Using  your subintervals,  find the lower sum and the upper sum  which approximate the definite integral from  0  to  1  of x3.

 

(d)  Same as (b) and (c)  except select  five intermediate points instead of three.  (Still keep the length of the longest subinterval  less that 0.3.)  Look at the graph of the function and choose the points in to get better approximations for your lower and upper sums then equally spaced intermediate points would give.