Math 414: Analysis I
Exam 2 (80 points)
Spring 2014
Name:
*No notes or electronic devices. You must show all your work to receive full credit.
When justifying your answers, use only those techniques that we learned in class.
1. (5) Let (an ) be a sequence of real numbers. Consider the following statements.
(a) (an ) is bounded.
(b) (an ) is Cauchy.
(c) (an ) is monotone.
(d) (an ) is convergent.
(e) (an ) has a convergent subsequence.
(f) Every subsequence of (an ) converges.
Based on the results we discussed in class, indicate at least five relations among the
above. For instance (d) implies (f) since every subsequence of a convergent sequence
is convergent.
1
Math 414: Analysis I
Exam 2 (80 points)
Spring 2014
2. Determine if the following statements are true or false. If the statement is true,
prove it. If the statement is false, provide a counterexample. (14 points total, 7
points each)
(a) Suppose that (xn ) is a sequence of real numbers such that |xn | ≤ n for all
n ∈ N, then (xn ) has a convergent subsequence.
(b) If (xn ) is a monotone increasing sequence of nonnegative real numbers, and
lim inf xn exist and is finite, then (xn ) converges.
n→∞
2
Math 414: Analysis I
Exam 2 (80 points)
Spring 2014
3. (a) (10) If an 6= 0 for all n ∈ N, lim an = 1 and lim bn does not exist, prove that
n→∞
n→∞
lim an bn does not exist. Suggestion: Use contradiction.
n→∞
(b) (10) Suppose that (an ) is a sequence of real numbers satisfying
|an | ≤
n+1
n
for all n ∈ N. Define bn := an cos(n3 ). Prove that (bn ) has a convergent
subsequence. Can we conclude that (bn ) converges? Why/Why not?
3
Math 414: Analysis I
Exam 2 (80 points)
Spring 2014
GROUP 1 PROBLEMS: Please pick ONE problem to complete.
4. (15) Suppose that x ∈ R. Prove that there exist a sequence of rational numbers
(rn ) such that lim(rn ) = x. (Hint: Recall the Density Theorem.)
5. (15) Suppose that (xn ) is a sequence of positive real numbers such that lim (x1/n
n ) =
n→∞
P∞
L < 1. Prove that i=1 xn converges.
4
Math 414: Analysis I
Exam 2 (80 points)
Spring 2014
GROUP 2 PROBLEMS: Please pick ONE problem to complete.
6. (a) (10) Does the following series converge or diverge?
∞
X
xn
where xn :=
n=1


1
2n
n is even

1
3n
n is odd
Justify your answer.
(b) (15) Let (xn ) and (yn ) be strictly positive sequences such that
lim
n→∞
xn
yn
= 1.
P
xn is convergent, prove that
yn is convergent.
√
7. (a) (10) Prove that the sequence ( n − 1) is properly divergent.
an
(b) (15) Prove that if lim
= L > 0, then lim an = +∞.
n→∞ n
n→∞
If
P
5