Convergence of Series
y=
2
1
x2
Z
1
12
1
1
1
22
1
UCSB Mathematics
2
k=1
1
32
3
X 1
1
dx
x2
k2
1
42
4
Convergence of Series
1
52
5
1
62
6
1
72
7
October 10, 2016
x
1/8
A Different Point of View
y=
2
1
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Matthew Choi
Assignment HW3 due 01/29/2015 at 11:59pm PST
Math6B-01-W15-CHEN
4. (1 pt) The following series are geometric series or a sum
of two geometric series.
Determine whether each series converg
Matthew Choi
Assignment HW4 due 02/05/2015 at 11:59pm PST
Math6B-01-W15-CHEN
1. (1 pt) Use the limit comparison test to determine whether
6n3 6n2 + 16
an = 6 + 3n4 converges or diverges.
n=16
n=16
1
Matthew Choi
Assignment HW7 due 02/26/2015 at 11:59pm PST
Math6B-01-W15-CHEN
(c) Which function given below could the Fourier approxi1. (1 pt) (a) Suppose youre given the following Fourier
mation F5 (
Practice Problems
Math 6B
Spring 2017
Z
(y + e
1. Compute the path integral
x
)dx + (2x + cos(y 2 )dy where ~c is the positively
~c
oriented boundary of the region enclosed by the parabolas y = x2 and
Practice Problems
Math 6B
Spring 2017
1. Determine whether the series converges. Please justify your answer.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
X
n=1
X
n=1
X
n=1
X
9
(1 + n)4/3
n 41n
n
2n
(ln n)2
n2
n=1
X
Practice Problems
Math 6B
Spring 2017
1. Determine whether the series converges. Please justify your answer.
(a)
X
n=1
9
(1 + n)4/3
Solution: This series is convergent. We know that the series
X
1
n4/
Practice Problems
Math 6B
1. Compute the path integral
Z
Spring 2017
(y + e
p
x
)dx + (2x + cos(y 2 )dy where ~c is the positively
~c
oriented boundary of the region enclosed by the parabolas y = x2 a
MATH 6BWinter 2016
Vector Calculus II
Study Guide Midterm 2: Extra Problems
1. Recognize a geometric series. In the following examples, give a check mark to the geometric series and
two check marks to
MATH 6BVector Calculus II, Spring 2016
Midterm Exam 1Solutions
Use a different page for each problem.
Show your work, otherwise no partial credit will be given.
No technology is permitted.
Submit
MATH 6BWinter 2016
Vector Calculus II
Study Guide Midterm 2: Extra Problems
1. Recognize a geometric series. In the following examples, give a check mark to the geometric series and
two check marks to
Practice Problems
Math 6B
Spring 2017
Z
(y + e
1. Compute the path integral
x
)dx + (2x + cos(y 2 )dy where ~c is the positively
~c
oriented boundary of the region enclosed by the parabolas y = x2 and
Practice Problems
Math 6B
Spring 2017
1. Determine whether the series converges. Please justify your answer.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
1
X
9
(1 + n)4/3
n=1
1
X
n 41
n=1
1
X
(ln n)2
n2
n=1
1
X
n
2n
Math 6B Midterm Review #1
1
Greens Theorem
(y + e
1. Compute the line integral
x
) dx + (2x + cos(y 2 ) dy where C is the positively oriented boundary
C
of the region D enclosed by the parabolas y = x
Math 6B
Solutions to Practice Problems for the final
Here are the solutions for practice problems for the final. All problems have appeared on
Math 6B exams in the past
1. Consider the curve C paramet
Math 6B
Practice Problems for Exam 2
Here are a list of practice problems for Exam 2. All problems have appeared on Math
6B exams in the past
1. Determine whether the following series converge absolut
Math 6B
Practice Problems for Exam 2
Here are a list of practice problems for Exam 2. All problems have appeared on Math
6B exams in the past
1. Determine whether the following series converge absolut
Clockwise/ inward = negative
Counter Clockwise/ outward = Positive
Change of Variables:
R
s
f ( x , y ) dA= f ( g (u , v ), h( u , v)
o dA =
| |
(x , y )
dudv
(u , v )
| |
(x, y)
dudv
(u , v )
(Jacob
Math 6B
Practice Problems for Exam 1
Here are a list of practice problems for Exam 1. All problems have appeared on Math
6B exams in the past
1. Integrate the vector field F(x, y, z) = i + z 2 j + (y
Clockwise/ inward = negative
Counter Clockwise/ outward = Positive
Change of Variables:
R
s
f ( x , y ) dA= f ( g (u , v ), h( u , v)
o dA =
| |
(x , y )
dudv
(u , v )
| |
(x, y)
dudv
(u , v )
(Jacob
FORMULAS
v= f
I
1
r2
=10 log 10
f beat = f 1 f 2
n =
1
9 N m
k=
9.0 10
4 0
C2
2L
n
E =
C=
2
Q
U=
2C
r
2
v=
V =k
q
r
4L
odd n
n
r
U =k
0 A
d
2 0
q 1 q2
r
V battery = Ir
2
V
P=
R
2
P=I R
1
1 1
R tot
1. Note this function is even, so its Fourier series contains only cosine terms.
Recall our formula for the coefficients:
Z
1
( |x|) cos kxdx
ak =
Z
2
=
( x) cos kx
0
Z
sin kx
( x)
2
= (
sin kx)
Using Power Series to Solve ODEs, Part III
Math 6B John Kaminsky
UCSB
11/6/2017
1/19
Where we are
We can use power series to solve ODEs.
2/19
Where we are
We can use power series to solve ODEs.
If a p
Using Power Series to Solve ODEs, Part II
Math 6B John Kaminsky
UCSB
11/3/2017
1/17
Example 1
Consider the differential equation
x2
d 2y
dy
+ 4x
+ 2y = 0.
2
dx
dx
2/17
Example 1
Consider the different
Using Power Series to Solve ODEs
Math 6B John Kaminsky
UCSB
11/1/2017
1/19
Last Time
Last class, we solved ODEs by assuming the solution is a
power series.
2/19
Last Time
Last class, we solved ODEs by
Power Series
Math 6B John Kaminsky
UCSB
10/25/2017
1/19
Taylor Polynomial
If f can be differentiated n times at x0 , then we define the n-th
degree Taylor polynomial for f about x = x0 to be
f 00 (x0
Series Part 2
Math 6B John Kaminsky
UCSB
10/23/2017
1/21
Example 2
Does the series
X
(12)n
k=1
n
converge?
2/21
Example 2
Does the series
X
(12)n
n
k=1
converge?
We have
1
= lim |un | n
n
1
(12)n n
Math 6B
HW8
Here are a couple of problems about PDEs for HW8. I expect you to at least attempt
each problem in order to receive full credit. I will post detailed solutions so you can check
your work a
Math 6B
HW8
Here are a couple of problems about PDEs for HW9. I expect you to at least attempt
each problem in order to receive full credit. I will post detailed solutions so you can check
your work a
Math 6B
HW7 Supplement
Here are a couple of supplemental problems about Fourier Series for HW7. I expect you
to at least attempt each problem in order to receive full credit. I will post detailed solu
Change of variables for double integrals
Change of variables for Triple Integrals
Vector Calculus II
Math 6B John Kaminsky
UCSB
9/29/2017
1/22
Change of variables for double integrals
Change of variab
Divergence Theorem
Math 6B John Kaminsky
UCSB
10/11/2017
1/22
Question to Ponder from Last Class
The flux of
F(x, y , z) = x 2 i
through any surface is positive.
2/22
Gradient
The gradient operator is