This is a list of some recommended review problems from the test.
I have tried to get a good sampling of topics from throughout the course, 
but don't exclusively focus on studying these problems (make sure to learn 
all definitions for example).

1 Ex 4.4 pg 19
2 Example 11.3 pg 151 has a formula for a_n on page 152.  Prove this formula using
induction.
3 We proved that each natural number is the product of primes using Strong Induction.
Prove that this decompositions is unique.
Hint: For the inductive step, assume that 
(p1 ^ s1) * (p2 ^ s2) * ... (pn ^ sn) = (q1 ^ t1) * ( q2 ^ t2) * ... * (qm ^ tm).
(Where the p's and q's are prime.  Note that this is just two decomposition into 
primes.  We want to prove that since they are equal, they are the same primes (after 
maybe reordering).
So the idea is to note that p1 divides both sides of the equation evenly, so that
p1 divides qi for some i.  Work from here and try to get an inductive proof.
4 Ex 1.8 e) and f) pg 31
5 Ex 2.14 pg 38
6 Let A, and B be intervals inside [0,1].  (An interval is something like (a,b) or [a,b) where
the first is all numbers x where a < x < b.  The second is all numbers x,  a < x < b plus the 
number a (so the first less than should be less than or equal to).
Assume the following are true A union B = [0,1] (all points of [0,1] are in A or B)
0 is in A.  1 is in B. sup A is not in A.  inf B is not in B.
Show that A intersect B is not empty (that is that there is a point x0 both in A and B).
Hint: break into three cases sup A < inf B, sup A = inf B, sup A > inf B.
7 Ex 1.14 pg 47
8 Ex 1.19 pg 48
9 Ex 1.22 pg 48
10 Ex 2.14 c,d,e pg 56
11 Ex 2.15 b,f pg 57
12 Ex 4.4 d pg 61
13 Ex 4.5 c,d pg 61
14 Ex 6.13 pg 70
15 Ex 7.3 pg 75
16 Ex 1.12 e,j pg 86
17 Ex 2.17 pg 94
18 Ex 5.12 c,h,o,u pg 113
19 Ex 6.13 pg 121
20 Ex 6.15 f pg 123
21 Ex 7.8 b,e,h,j pg 128
22 Ex 8.15 pg 135
23 Ex 8.17 a pg 136
24 Ex 9.5 a pg 142
25 In addition to normal sin and cosine functions there are hyperbolic versions sinh and cosh
that can be defined as follows:
sinh x = (e^x - e^-x) / 2
cosh x = (e^x + e^-x) / 2
Simple examination shows the following formulas,
(d/dx) (sinh x) = cosh x
(d/dx) (cosh x) = sinh x
Derive a power series expansion for sinh x in two ways: 
i) use the power series expansion that we already know for e^x.
ii) Use Taylor's Theorem to show that a power series expansion for sinh x exists. 
Then, knowing that a power series exists, solve for it.  
(In this part, don't use the power series for e^x, that was the method for part i).
Hint: Unlike sin and cos, sinh and cosh are not bounded by some constant value (both
go to infinity as x goes to infinity).  However, look at the note at the end of 
Theorem 9.1 on page 141 (and the following example).
26 Ex 10.5 pg 146
27 11.5 c pg 153