John Zhu

John Zhu

Absolute Convergence

Slide Duration:

Table of Contents

Section 1: Prerequisites
Parametric Curves

10m 10s

Intro
0:00
Parametric Equations
0:23
Familiar Functions
0:32
Parametric Equation/ Function
0:56
Example 1: Graph Parametric Equation
1:48
Example 2
4:30
Example 3
6:01
Example 4
7:12
Example 5
8:10
Polar Coordinates

14m 54s

Intro
0:00
Polar Coordinates
0:09
Definition
0:11
Example
0:23
Converting Polar Coordinates
1:49
Example: Convert Polar Equation to Cartesian Coordinates
3:06
Example: Convert Polar Equation to Cartesian Coordinates
5:21
Example: Find the Polar Representation
6:51
Example: Find the Polar Representation
8:39
Example: Sketch the Graph
10:02
Vector Functions

12m 5s

Intro
0:00
Vector Functions
0:11
2 Parts
0:18
Example 1
1:44
Example 2
3:52
Example 3
4:59
Example 4
6:03
Example 5
8:00
Example 6
9:23
Example 7
10:16
Section 2: Differentiation
Parametric Differentiation

13m 15s

Intro
0:00
Parametric Differentiation
0:15
Example 1
1:16
Example 2
1:54
Example 3
3:15
Example 4
4:59
Parametric Differentiation: Position, Speed & Acceleration
7:45
Example 5
8:32
Example 6
10:37
Polar Differentiation

10m 14s

Intro
0:00
Polar Differentiation
0:11
Goal
0:13
Method
0:24
Example 1
0:52
Example 2
4:27
Example 3
7:03
L'Hopital's Rule

6m 43s

Intro
0:00
L'Hopitals Rule
0:09
Conditions
0:14
Limit
0:36
Example 1
1:21
Example 2
2:38
Example 3
4:09
Example 4
5:01
Section 3: Integration
Integration By Parts

19m 4s

Intro
0:00
Integration by Parts
0:12
Formula
0:31
How It's Derived From Product Rule
1:09
Integration by Parts: Rules to Follow
3:24
Recap of the Rules
4:13
Example 1
4:29
Example 2
7:04
Example 3
8:40
Example 4
13:48
Example 5
15:36
Integration By Partial Fractions

22m 4s

Intro
0:00
Integration by Partial Fractions: Goal
0:08
Example 1
2:10
Example 2
7:10
Example 3
10:06
Example 4
14:23
Example 5
16:26
Improper Integrals

19m 1s

Intro
0:00
Improper Integrals
0:06
3 Steps
1:11
Example 1
1:46
Example 2
3:20
Example 3
6:05
Example 4
9:02
Example 5
11:21
Example 6
15:54
Section 4: Applications of Integration
Logistic Growth

19m 16s

Intro
0:00
Logistic Growth Function
0:07
Defining Variables
0:40
Equation Parts
1:51
Logistic Growth
2:04
Example 1
2:59
Example 2
7:13
Example 3
11:29
Example 4
15:21
Arc Length for Parametric & Polar Curves

17m 40s

Intro
0:00
Arc Length
0:13
Arc Length of a Normal Function
0:24
Example
1:27
Example 2: Arc Length for Parametric Curves
3:31
Example 3: Arc Length for Parametric Curves
4:23
Example 4: Arc Length for Parametric Curves
8:05
Example 5: Arc Length for Parametric Curves
12:22
Example 6: Arc Length for Polar Curves
15:36
Example 7: Arc Length for Polar Curves
16:03
Area for Parametric & Polar Curves

4m 33s

Intro
0:00
Area for Parametric Curves: Parametric Function
0:10
Example 1: Area for Parametric Curves
0:35
Area for Parametric Curves: Polar Function
2:50
Example 2: Area for Polar Curves
3:18
Section 5: Sequences, Series, & Approximations
Definition & Convergence

7m 10s

Intro
0:00
Sequences: Definition
0:09
Definition
0:31
Example 1
1:07
Sequences: Convergence
2:02
Example 1
2:52
Example 2
3:36
Example 3
4:47
Example 4
6:16
Geometric Series

10m 59s

Intro
0:00
Geometric Series
0:07
Definition
0:24
Expanded Form
0:39
Convergence Rules
1:00
Example 1: Convergence
1:22
Example 2: Convergence
2:36
Example 3: Convergence
3:45
Sum of Series
5:04
Sum of First n Terms
5:14
Sum of Series
5:24
Example 1: Sum of Series
5:39
Example 2: Sum of Series
7:15
Example 3: Sum of Series
8:24
Example 4: Sum of Series
9:36
Harmonic & P Series

3m 58s

Intro
0:00
Harmonic Series
0:08
P-Series
1:17
Example 1: P-Series Test
2:22
Example 2: P-Series Test
3:06
Integral Test

11m 31s

Intro
0:00
Integral Test
0:09
Example 1
0:54
Example 2
4:00
Example 3
7:59
Example 4
9:47
Comparison Test

7m 47s

Intro
0:00
Comparison Test
0:10
Example 1
1:07
Example 2
2:33
Example 3
4:20
Example 4
6:29
Limit Comparison Test

9m 40s

Intro
0:00
Limit Comparison Test
0:09
Conditions
0:31
Example 1
1:01
Example 2
2:53
Example 3
4:15
Example 4
6:19
Ratio Test

11m 20s

Intro
0:00
Ratio Test
0:09
Example 1
0:57
Example 2
2:55
Example 3
6:27
Example 4
8:36
Alternating Series

9m 11s

Intro
0:00
Alternating Series
0:08
Convergence Test
0:59
Example 1
1:27
Example 2
2:42
Example 3
4:57
Example 4
6:37
Absolute Convergence

7m 13s

Intro
0:00
Absolute Convergence
0:12
Example 1
0:52
Example 2
3:42
Example 3
5:21
Power Series Convergence

12m 52s

Intro
0:00
Power Series
0:09
Definition
0:19
Radius & Interval of Convergence
2:07
Example 1
2:28
Example 2
4:18
Example 3
6:20
Example 4
9:11
Taylor Series

15m 11s

Intro
0:00
Taylor Series
0:06
MacLaurin Series
0:45
Example 1
1:02
Example 2
2:45
Example 3
6:04
Example 4
8:23
Example 5
11:49
Power Series Operations

16m 40s

Intro
0:00
Operations
0:07
Example 1: Substitution
1:05
Example 2: Substitution
3:41
Example 3: Differentiation/ Integration
5:39
Example 4: Differentiation/ Integration
9:55
Example 5: Multiplying/ Dividing
12:32
Example 6: Multiplying/ Dividing
14:50
Lagrange Error

7m 9s

Intro
0:00
Power Series: Lagrange Error
0:06
Lagrange Remainder
0:21
Lagrange Error Bound
0:50
Example 1
1:06
Example 2
3:27
Section 6: Practice Test
AP Calc BC Practice Test Section 1: Multi Choice Part 1

15m 45s

Intro
0:00
Practice Problem 1
0:10
Practice Problem 2
1:19
Practice Problem 3
2:33
Practice Problem 4
5:25
Practice Problem 5
6:32
Practice Problem 6
9:11
Practice Problem 7
11:31
Practice Problem 8
13:08
Practice Problem 9
14:16
AP Calc BC Practice Test Section 1: Multi Choice Part 2

21m 38s

Intro
0:00
Practice Problem 10
0:12
Practice Problem 11
3:03
Practice Problem 12
4:53
Practice Problem 13
6:56
Practice Problem 14
9:10
Practice Problem 15
12:17
Practice Problem 16
14:43
Practice Problem 17
16:18
Practice Problem 18
18:34
AP Calc BC Practice Test Section 1: Multi Choice Part 3

17m 31s

Intro
0:00
Practice Problem 19
0:09
Practice Problem 20
3:03
Practice Problem 21
4:07
Practice Problem 22
5:41
Practice Problem 23
8:48
Practice Problem 24
11:16
Practice Problem 25
12:53
Practice Problem 26
14:12
Practice Problem 27
15:21
AP Calc BC Practice Test Section 1: Multi Choice Part 4

14m 17s

Intro
0:00
Practice Problem 28
0:08
Practice Problem 29
1:19
Practice Problem 30
2:15
Practice Problem 31
4:19
Practice Problem 32
6:54
Practice Problem 33
8:13
Practice Problem 34
9:03
Practice Problem 35
10:21
Practice Problem 36
11:42
AP Calc BC Practice Test Section 1: Multi Choice Part 5

22m 45s

Intro
0:00
Practice Problem 37
0:10
Practice Problem 38
4:09
Practice Problem 39
7:21
Practice Problem 40
9:42
Practice Problem 41
11:35
Practice Problem 42
14:01
Practice Problem 43
15:50
Practice Problem 44
16:43
Practice Problem 45
19:35
AP Calc BC Practice Test Section 1: Free Response Part 1

9m 55s

Intro
0:00
Practice Problem 1a
0:08
Practice Problem 1b
1:30
Practice Problem 1c
3:08
Practice Problem 2a
3:50
Practice Problem 2b
5:40
Practice Problem 2c
7:24
AP Calc BC Practice Test Section 1: Free Response Part 2

10m 8s

Intro
0:00
Practice Problem 3a
0:09
Practice Problem 3b
3:01
Practice Problem 4a
3:55
Practice Problem 4b
5:43
AP Calc BC Practice Test Section 1: Free Response Part 3

13m 58s

Intro
0:00
Practice Problem 5a
0:07
Practice Problem 5b
2:21
Practice Problem 5c
4:21
Practice Problem 6a
6:16
Practice Problem 6b
8:57
Practice Problem 6c
11:03
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Absolute Convergence

  • If converges then “absolutely converges”
  • Convergence check
    • Test for absolute convergence
    • If not absolutely convergent, test for conditional convergence or divergence

Absolute Convergence

Does the series ∑n = 1 ( − 1)n + 1[1/(n2)] converge absolutely?
  • Determine if ∑n = 1 | an | converge
  • n = 1 | ( − 1)n + 1[1/(n2)] | = ∑n = 1 | [1/(n2)] |
Apply P - series properties
n = 1 | [1/(n2)] | = 0 Thus the series converges completely
Does the series ∑n = 1 ( − 1)n + 1[(n − 1)/(n2 + 2)] converge absolutely?
  • Determine if ∑n = 1 | an | converge
  • n = 1 | ( − 1)n + 1[(n − 1)/(n2 + 2)] | = ∑n = 1 | [(n − 1)/(n2 + 2)] |
  • Apply Harmonic series properties
= [1/n] = diverges
Thus the series does not absolutely converge
Does the series ∑n = 1 [(( − 1)n + 1)/(√n + 3)] absolutely converge?
  • Determine if ∑n = 1 | an | converge
  • n = 1 | [(( − 1)n + 1)/(√n + 3)] | = ∑n = 1 | [1/(√n + 3)] |
  • Apply Comparison Test
  • n = 1 | [1/(√n + 3)] | compared with ∑n = 1 | [1/(√n )] |
  • Apply P - Series properties
  • p = [1/2] < 1 = diverges
Thus the series does not absolutely converge
Does the series ∑n = 1 [(( − 1)n)/(3n − 1)] absolutely converge?
  • Determine if ∑n = 1 | an | converge
  • n = 1 | [(( − 1)n)/(3n − 1)] | = ∑n = 1 | [1/(3n − 1)] |
a = 1r = [1/3]|r|< 1, thus the series absolutely converges
Does the series ∑n = 1 [(( − 1)n + 1)/(4√{n + 2})] absolutely converge?
  • Determine if ∑n = 1 | an | converge
  • n = 1 | [(( − 1)n + 1)/(4√{n + 2})] | = ∑n = 1 | [1/(4√{n + 2})] |
n = 1 | [1/(4√{n + 2})] | = ∑n = 1 | [1/(( n + 2 )1/4)] | p = [1/4] < 1 = diverges
Thus the series does not absolutely converge
Does the series ∑n = 1 [(( − 1)n + 1n2)/(3√{n})] absolutely converge?
  • Determine if ∑n = 1 | an | converge
= ∑n = 1 | n5/3 | = ∞ Thus the series does not absolutely converge
Does the series ∑n = 1 [(2( − 1)n(n + 2)2)/(n2 + 1)] absolutely converge?
  • Determine if ∑n = 1 | an | converge
  • n = 1 | [(2( − 1)n(n + 2)2)/(n2 + 1)] | = ∑n = 1 | [(2(n + 2)2)/(n2 + 1)] |
  • = ∑n = 1 | [(2(n2 + 2n + 4))/(n2 + 1)] |
  • = ∑n = 1 | [(2n2 + 4n + 8)/(n2 + 1)] |
  • = [(2n2)/(n2)]
  • = 2
Thus the series does absolutely converges
Does the series ∑n = 1 [(( − 1)n)/(3n + 1)] absolutely converge?
  • Determine if ∑n = 1 | an | converge
  • n = 1 | [(( − 1)n)/(3n + 1)] | = ∑n = 1 | [1/(3n + 1)] |
Let bn = [1/n]limn∞ [([1/(3n + 1)])/([1/n])] = [1/3] Thus the series absolutely converges
Does the series ∑n = 1 [(( − 1)n + 2(n − 1))/(n3)] absolutely converge?
  • Determine if ∑n = 1 | an | converge
  • n = 1 | [(( − 1)n + 2(n − 1))/(n3)] | = ∑n = 1 | [(n − 1)/(n3)] |
Let bn = [1/(n2)]limn∞ [([(n − 1)/(n3)])/([1/(n2)])] = 1 Thus the series absolutely converges
Does the series ∑n = 1 [(( − 1)n + 12n)/n!] absolutely converge?
  • Determine if ∑n = 1 | an | converge
  • n = 1 | [(( − 1)n + 12n)/n!] | = ∑n = 1 | [(2n)/n!] |
Apply the Ratio Test
n = 1 | [(2n)/n!] | = 0 Thus the series does absolutely converges

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Absolute Convergence

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  • Intro 0:00
  • Absolute Convergence 0:12
  • Example 1 0:52
  • Example 2 3:42
  • Example 3 5:21
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