MATH 407 Summer 2021 Calculus Final ExamMATH 407 Summer 2021 Final Exam – June 24, 2021

Due 11:45 am. Late submissions will NOT be accepted!

1 (20 pts.) Answer TRUE or FALSE to each statement. No explanation is required. Each part

is worth 2 pts.

(a) The sequence sn = (−1)n

1 +

1

n

has a convergent subsequence.

(b) If S and T are nonempty bounded subsets of R such that S ⊂ T, then inf S < inf T.
(c) A sequence (sn) converges to s if and only if lim inf
n→∞
sn = lim sup
n→∞
.
(d) The series X cos(nx)

n2

defines a continuous function on R.

(e) If f is continuous at x = a, then f is differentiable at x = a.

(f) If (sn) is a monotone and bounded sequence, then it is also Cauchy.

(g) The function f(x) = 1

x

2

is uniformly continuous on [−2, 2].

(h) The sequence of functions fn(x) = x

n

converges uniformly to the function f(x) = 0 on

[0, 1].

(i) If Z b

ag(x)

2

dx = 0, then g(x) = 0 for all x ∈ [a, b].

(j) If Z b

a

g(x) dx = 0, then g(x) = 0 for all x ∈ [a, b]

2 (20 pts.) Define a sequence (sn) by iteration:

s1 = 0 and sn+1 =

√

sn + 1.

(a) (5 pts.) Use induction to show that sn+1 − sn > 0 for all n ∈ N.

(b) (5 pts.) Use induction to show that sn 6 3 for all n ∈ N.

(c) (5 pts.) Using parts (a) and (b) explain why (sn) converges.

(d) (5 pts.) Find the limit of (sn). (Hint: If sn → s then also sn+1 → s.)

3 (10 pts.) This problem consists of five parts. Make sure that you justify your answer!

(a) (2 pts.) Give an example of a divergent series Pan such that the series Pa

2

n

converges.

(b) (2 pts.) Give an example of a convergent series Pan such that the series Pa

2

n diverges.

(c) (2 pts.) Give examples of a convergent series Pan and a divergent series P

P

bn such that

anbn converges.

(d) (2 pts.) Give examples of a convergent series Pan and a divergent series P

P

bn such that

anbn diverges.

(e) (2 pts.) Give examples of a convergent series Pan and a convergent series P

P

bn such that

anbn diverges

4 (15 pts.) Consider the sequence of functions

fn(x) = nx

1 + n2x

2

defined on the whole real line R.

(a) (5 pts.) Find the function f such that fn → f pointwise as n → ∞.

(b) (10 pts.) Does fn converge to f from part (a) uniformly? Prove your answer.

5 (15 pts.) Consider the function f on R given by the formula

f(x) = (

x

2

if x > 1,

x if x < 1.
(a) (5 pts.) Prove that the function f is continuous at x = 1.
(b) (10 pts.) Prove that the function f is not differentiable at x = 1.
Pick any TWO of the following five problems:
6 (10 pts.) Consider the function
f(x) = x
x + 2
.
(a) (6 pts.) Prove that the function f is uniformly continuous on the interval (2, 4).
(b) (4 pts.) Is the function f uniformly continuous on the interval (−4, 4). Justify your
answer.
7 (10 pts.) This problem consists of two parts.
(a) (2 pts.) Write down the definition of limn→∞
sn = s.
(b) (8 pts.) Using the definition in part (a) show that limn→∞
2n + 5
5n − 9
=
3
4
.
8 (10 pts.) Show that the function f(x) = 1
x
is integrable on [1, 5].
9 (10 pts.) Suppose f is a differentiable function on R such that
|f
0
(x)| 6 28 ∀ x ∈ R.
Prove that f is uniformly continuous on R. (Hint: Use the Mean Value Theorem.)
10 (10 pts.) Show that the function
f(x) = (
x if x ∈ [0, 1] ∩ Q,
0 if x ∈ [0, 1] ∩ (RQ).
is not integrable on R.

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