Math 414: Analysis I
Practice Exam 2
Spring 2014
Name:
*Please be aware that this practice test is longer than the test you will see on March 14,
2014. Also, this test does not cover every possible topic that you are responsible for on
the exam. For a comprehensive list of all topics covered on the exam, please see the exam
topics document on the website. The best way to prepare for the exam is to go back over
your notes, noting every theorem/proposition/lemma done in class, redoing examples in
the notes, going through your homework assignments and practicing the extra practice
problems.
1. For this problem, let (xn ) be a sequence of real numbers such that lim xn = x.
n→∞
(a) Suppose that xn > a for all n ∈ N. Prove that x ≥ a.
(b) Construct an example to show that if xn > a then it is not necessarily true
that x > a.
2. Suppose that (cn ) is a sequence of real numbers such that
|cn | ≤
n+1
n3
for all n ∈ N. Prove that xn is a Cauchy sequence.
3. Suppose that (xn ) and (yn ) are sequences of real numbers such that lim(xn ) exist
and lim(xn + yn ) exist. Prove that (yn ) converges.
4. If a, b > 0, show that
o a+b
np
(n + a)(n + b) − n =
n→∞
2
lim
5. Consider the recursively defined sequence:
x1 := 8,
xn+1 :=
xn
+2
2
Prove that this sequence converges, and find the limit.
6. Using only the definition of convergence, prove that lim 2 +
(−1)n
n
= 2.
7. Determine if the following statements are true or false. If the statement is true,
prove it. If the statement is false, provide a counterexample (if possible) or explain
why the claim is false.
(a) If (xn ) and (yn ) are sequences of real numbers such that lim(xn + yn ) exist,
then lim(xn ) and lim(yn ) exist.
(b) If (xn ) is a convergent sequence of real numbers, then any subsequence of (xn )
is Cauchy.
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Math 414: Analysis I
Practice Exam 2
Spring 2014
(c) If x1 := 1 and xn+1 := x2n + xn , then lim(xn ) = 0.
(d) If (an ) is a sequence of real numbers such that lim(an ) = 0, and (xn ) is a
sequence of real numbers such that for some x ∈ R we have
|xn − x| ≤ an
(1)
for all n ∈ N, then (xn ) converges and lim xn = x.
(e) Suppose that (an ) is a sequence such that every subsequence of (an ) converges
to some a ∈ R. Then (an ) converges to a.
(f) If lim(an+1 − an ) = 0 for every k ∈ N then (an ) converges.
(g) If (xn ) is unbounded, then (xn ) is properly divergent.
8. Determine if the following series converges or diverges. Justify your answer.
∞
X
n=1
n3
n
√
+n+ n+1
9. Let (xn ) and (yn ) be sequences of real numbers. Suppose that
• limn→∞ xn = ∞.
• There is a number M > 0 such that 0 < yn < M for every n.
Prove that
xn
= ∞.
n→∞ yn
lim
10. Prove the following part of the Limit Comparison Test: Let (xn ) and (yn ) be strictly
positive sequences such that
lim
n→∞
If
P
yn is convergent, prove that
P
xn
yn
= 0.
xn is convergent.
11. Suppose that (xn ) is a bounded sequence and (xnk ) is any subsequence of (xn ).
(a) Prove that lim sup xnk ≤ lim sup xn .
n→∞
n→∞
(b) Determine a sufficient condition on (xn ) so that lim sup xnk = lim sup xn .
n→∞
n→∞
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