Math 1320.001 Exam 02 Homework Assignment
Fall 2015
Name
Student ID Number:
Class Number:
Instructions:
• This homework assignment is due: Tuesday, 17 November 2015 at 9:25am
• Before submitting this “exam”, photocopy your work and keep
the photocopied original for a future assignment. The ORIGINAL should be submitted with your homework.
• Work the problems on scratch paper and then copy your polished solution
into a printed copy of this “exam”.
• Your work should be clear and complete. For example, this means that
– Calculations involving limits should have proper limit notation
– Calculations involving L’hopital’s rule should use a corresponding
function and mention the type of indeterminate form
– Claims of convergence/divergence should include a description of
any series tests that are being used and all the calculations necessary
to show that the test applies.
– Claims of convergence/divergence should include the names of any
classic series/sequences that are used as well as descriptions of why
the classic example applies.
Math1320.001
Exam 02 – Hwk Assignment, Page 2 of 12 Due: 17 November 2015
For Grader Use Only
Question Points Score
1
9
2
18
3
6
4
12
5
12
6
18
7
6
8
6
9
18
Total:
105
Math1320.001
Exam 02 – Hwk Assignment, Page 3 of 12 Due: 17 November 2015
WRITE YOUR ANSWERS IN THE PROVIDED BOXES
SHOW YOUR WORK
[9 points] 1. Determine whether each of the following sequences converges. If it does, find the limit
of the sequence. Be sure to show you work and give the name of any tests or classic
sequences you use.
(a) {an } =
2+n3
1+2n3
(b) {an } = 6 − (0.7)n
(c) {an } =
sin(nπ)
√
n
Math1320.001
Exam 02 – Hwk Assignment, Page 4 of 12 Due: 17 November 2015
[18 points] 2. Determine whether the following series converge. You do not need to find the sum. Be
sure to show your work and name any tests or classic sequences you use.
(a)
∞
X
n!
100n
n=1
(b)
∞
X
n(n + 2)
n=1
(n + 3)2
Math1320.001
Exam 02 – Hwk Assignment, Page 5 of 12 Due: 17 November 2015
(c)
∞
X
n=2
(d)
n3
n4 − 1
√
∞
X
(−1)(n−1) n
n=1
n+1
Math1320.001
Exam 02 – Hwk Assignment, Page 6 of 12 Due: 17 November 2015
[6 points] 3. Integral Test
(a) Use the Integral Test to show that the following p-series converges. Be sure to show
that the conditions of the Integral Test are satisfied.
∞
X
1
n4
n=1
(b) Using the Integral Test Remainder Estimate Theorem, find the bounds on the error
that results when the first term of the sum in part (a) is used as an estimate of
the entire sum.
Math1320.001
Exam 02 – Hwk Assignment, Page 7 of 12 Due: 17 November 2015
[12 points] 4. Power Series
(a) Express
1
2+x
as a power series.
(b) What is the radius of convergence for the series in part(a)?
Math1320.001
Exam 02 – Hwk Assignment, Page 8 of 12 Due: 17 November 2015
[12 points] 5. Using the Ratio Test, find the interval of convergence for
f (x) =
∞
X
xn
n=1
n2
Math1320.001
Exam 02 – Hwk Assignment, Page 9 of 12 Due: 17 November 2015
[18 points] 6. Given the point P (0, 1, 0), Q(2, 1, −1) and the vector v =< 1, 2, 3 >
(a) find a non-zero vector that is orthogonal (perpendicular) to the plane that containing the points P and Q and the vector v.
(b) Write an equation for this plane. Do not simplify.
(c) Write an equation for a line that lies on this plane in parametric form.
Math1320.001
Exam 02 – Hwk Assignment, Page 10 of 12Due: 17 November 2015
[6 points] 7. Suppose we have two vectors v1 and v2 and v3 .
(a) If v1 · v2 = 0 what does this tell us about the vectors?
(b) If v1 × v2 = 0 what does this tell us about the vectors?
(c) If v1 · (v2 × v3 ) = 0 what does this tell us about the vectors?
[6 points] 8. Find the angle between the planes described by the following equations
−x + y + z = 1
x − 2y + 3z = 1
(1)
(2)
Math1320.001
Exam 02 – Hwk Assignment, Page 11 of 12Due: 17 November 2015
[18 points] 9. (a) Using the definition of Taylor series, write the first 4 non-zero terms of the Taylor
expansion (i.e. polynomial) of sin( Π4 x) about x = 1.
Math1320.001
Exam 02 – Hwk Assignment, Page 12 of 12Due: 17 November 2015
(b) Using the Taylor Inequality Theorem, calculate the bound on the error that results
from using the 2rd degree Taylor Polynomial to approximate sin( Π4 x) for |x−1| ≤ 1.