MA 306 Midterm Exam
January 16, 2004
NAME:
28 pts 1. Complete the following with the definition or statement.
a) A set A is countable if
b) lim an = a means (give the − N definition)
n→∞
c) s = sup(A) if (give the two conditions in the definition)
d) x is a limit point of a set if (give the definition)
e) State the Axiom of Completeness.
f) A set O is open if (give the definition)
g) State the Heine-Borel Theorem.
28 pts 2. For each of the following, if the statement is true, simply write T to the left of the statement.
If the statement is false, write F to the left of the statement and give an example to illustrate
why the statement is false.
a) The union of an arbitrary collection of open sets is open.
b) The intersection of an arbitrary collection of compact sets is compact.
c) If F1 ⊇ F2 ⊇ F3 ⊇ · · · is a nested sequence of nonempty closed sets, then the intersection
∞
∩ Fn 6= ∅.
n=1
d) A countable set is closed.
e) If lim |an | = |a|, then lim an = a.
n→∞
n→∞
f) Every sequence has a convergent subsequence.
g) If lim an = 0, then
n→∞
∞
X
n=1
an is convergent.
12 pts 3. Prove lim a2n = a2 given that lim an = a. You must use an − N argument.
n→∞
n→∞
10 pts 4. Prove that if a set K is closed and bounded, then K is compact.
12 pts 5. Determine the convergence or divergence of the following three infinite series. Simply state
AC, CC, or D and the test which gives the result.
∞
X
(−1)n
√
a)
n
n=1
b)
∞
X
2n − 1
n=1
c)
5n + 7
∞
X
(−1)n
n=1
n3
10 pts 6. If an ≥ 0 and
∞
X
5 pts Bonus. If an ≥ 0 and
∞
X
an is convergent, prove that
n=1
n=1
∞
X
a2n is convergent.
n=1
an is convergent, prove that
∞ √
X
an
n=1
n
is convergent.