Math 166Z Quiz 4
Name:
1. Consider the following sequences:
(
0, n odd
an = 1
, n even
n
(
1, n odd
bn = 1
, n even
n
(a) Graph each of these sequences below
(b) Based on the graph, does an converge? If it converges, what is its limit?
(c) Does bn converge? If so, what is its limit?
2. Consider the sequence { 14 ,
1
4
+
1
42
,
1
4
+
1
42
+
1
43
,
1
4
+
1
42
+
1
43
+
1
44
, ...}
(a) Write out the first 5 terms of this sequence as decimals. Does this sequence appear
to converge? If so, what does it look like it is converging to? Try writing out a few
more terms if you aren’t sure.
(b) Consider the square above with sides of length one. What is the area of this square?
(c) Assuming the square is cut up evenly, what is the area of the largest black square
(the one in the lower left?
(d) What is the area of the next smallest black square?
(e) Write an expression for the area of the largest 5 black squares added together. Does
this look familiar?
(f) Write a similar expression for the area of the grey squares and the white squares.
(g) Use this to show that the sequence { 14 ,
converges to your guess in part (a).
1
4
+
1
42
,
1
4
+
1
42
+
1
43
,
1
4
+
1
42
+
1
43
+
1
44
, ...}