Question

1) (a) Determine if the following statements are true or false. If true give a reason or cite a theorem and if false, give a counterexample.

i) If { a n } is bounded, then it converges.

ii) If { a n } is not bounded, then it diverges.

iii) If { a n } diverges, then it is not bounded.

(b) Give an example of *divergent* sequences { a n } and
{ b n } such that { a n + b n } converges.

Answer #1

True or false (give a reason if true or a counterexample if
false):
(a) If u is perpendicular (in three dimensions) to v and w,
those vectors v and w are parallel. "
(b) If u is perpendicular to v and w, then u is perpendicular to
v + 2 w,
(c) If u and v are perpendicular unit vectors then II u - v" =
,.,fi,

Determine if each of the following statements is true or false.
If a statement is true, then write a formal proof of that
statement, and if it is false, then provide a counterexample that
shows its false.
1) For each integer a there exists an integer
n such that a divides (8n +7) and
a divides (4n+1), then a divides 5.
2)For each integer n if n is odd, then 8
divides (n4+4n2+11).

5. Indicate which of the following statements are true or false.
If false, please give a justification.
(a) The central limit theorem allows you to assume that your
data is generated from a Normal distribution.
(b) The accuracy of the central limit theorem approximation
increases as n tends to infinity.
(c) The accuracy of the central limit theorem approximation
increases as the standard error decreases.
(d) The central limit theorem cannot be applied to discrete
(integer) valued data.

For each of the following statements decide whether they are
True or False and give a short argument if True, or counter example
if False.
(1) ∀n ∈ Z, ∃m ∈ Z, n + m ≡ 1 mod 2.
(2) ∀n ∈ Z, ∃m ∈ Z, (2n + 1)^2 = 2m − 1.
(3) ∃n ∈ Z, ∀m > n, m^2 > 100m.

Determine if each of the following statements is true or false.
If it’s true, explain why. If it’s false explain why not, or simply
give an example demonstrating why it’s false
(a) If λ=0 is not an eigenvalue of A, then the columns of A fo
ma basis of R^n.
(b) If u, v ∈ R^3 are orthogonal, then the set {u, u − 3v} is
orthogonal.
(c) If S1 is an orthogonal set and S2 is an orthogonal set...

Determine if the following statements are true or false. If it
is true, explain why. If it is false, provide an example.
a.) If a and b are positive numbers, then (a+b)^x=a^x+b^x
b.) If x < y, then e^x < e^y
c.) If 0 < b <1 and x < y then b^x > b^y
d.) if e^(kx) > 1, then k > 0 and x >0

Determine whether the following sequences converge or diverge.
If a sequence converges, find its limit. If a sequence diverges,
explain why.
(a) an = ((-1)nn)/
(n+sqrt(n))
(b) an = (sin(3n))/(1- sqrt(n))

1)
For each of the following state whether it is true or false. If
true give one sentence exolanation. If false, give counterexample.
a)If f is differentiable on (0,1) and f is increasing on (0,1)
then f' > 0 on (0,1)
b)Suppose that f is thrice differentiable function defined on
(-1,1). Suppose that the second order Taylor polynomial of f at 0
is 1-x^2. Then f has a local extremum at 0.

State if true or false. If false, provide an explanation or
counterexample.
a) The series ∑n=1oo (−1)?
[(?4 2?)/3n] converges
absolutely.
b) The radius of convergence of the power series ∑
(0.25)? √(?)?? is 4.

Determine if each of the following statements is true or false.
If it’s true, explain why. If it’s false explain why not, or simply
give an example demonstrating why it’s false. (A correct choice of
“T/F” with no explanation will not receive any credit.)
(a) If two lines are contained in the same plane, then they must
intersect.
(c) Let n1 and n2 be normal vectors for two planes P1, P2. If n1
and n2 are orthogonal, then the planes...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 4 minutes ago

asked 16 minutes ago

asked 20 minutes ago

asked 24 minutes ago

asked 32 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago