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Lecture Comments (5)

0 answers

Post by Thomas Zhang on March 19, 2014

For example 5, doesn't the limit not exist because it's +inf x-->0+ and it's -inf when x-->=-

0 answers

Post by Akshay Tiwary on February 28, 2013

In the third problem, it would be simpler to substitute '-x' in the Maclauren Expansion for 1/1-x

2 answers

Last reply by: Acme Wang
Sun Jul 10, 2016 3:25 AM

Post by Troy Wingert on February 27, 2013

Just a comment about Example 1 on Power Series Operations .... there's a mistake in the solution....it should be 1 - x^2 + x^4 - x^6 ..... Threw me off for a bit.

Power Series Operations

  • Operating on a power series:
    • For substitution:
      • Find similar and simpler form of function
      • Find series of simpler form
      • Substitute appropriate term
    • For derivatives and integrals:
      • Find integral of function
      • Find series of integral
      • Differentiate each term in series
    • For multiplication and division
      • Find series of known portion
      • Multiply or divide by other variable portion

Power Series Operations

Find the MacLaurin Series for f(x) = [1/(1 + x3)]
Substitute ( − x3) for the MacLaurin series for [1/(1 − x)][1/(1 − x)] = ∑n = 0∞ (x)n [1/(1 − ( − x3))] = ∑n = 0∞ ( − x3)n = 1 − x3 + x6 − x9 + ...
Find the MacLaurin Series for f(x) = [1/(1 + 3x4)]
Substitute ( − x2) for the MacLaurin series for [1/(1 − x)][1/(1 − x)] = ∑n = 0∞ (x)n [1/(1 − ( − 3x4))] = ∑n = 0∞ ( − 3x4)n = 1 − 3x4 + 9x8 − 27x12 + ...
Find the first 3 non - zero terms of the MacLaurin Series for f(x) = cos(2x)
Substitute (2x) for the MacLaurin series for cosxcosx = ∑n = 0∞ [(( − 1)nx2n)/(2n)!] cos2x = ∑n = 0∞ [(( − 1)n(2x)2n)/(2n)!] = 1 − 2x2 + [2x/3]4 − [4x/45]6 + ...
Find the first 3 non - zero terms of the MacLaurin Series for f(x) = sin( − 3x)
Substitute ( − 3x) for the MacLaurin series for sin xsinx = ∑n = 0∞ [(( − 1)n − 1x2n − 1)/((2n − 1)!)] sin( − 3x) = ∑n = 0∞ [(( − 1)n − 1( − 3x)2n − 1)/((2n − 1)!)] = − 3x + [(9x3)/2] − [(81x5)/40] + ...
Use a series to solve limx→0 [sinx/x]
Use the MacLaurin Series of sinx to divide
limx → 0 [sinx/x] = limx → 0 [(∑n = 0∞ [(( − 1)n − 1x2n − 1)/((2n − 1)!)] )/x] = limx → 0 [(x − [(x3)/6] + [(x5)/120] − [x7/5040])/x] = limx → 0 1 − [(x3)/6x] + [(x5)/120x] − [(x7)/5040x] = 1 − 0 + 0 − 0 = 1
Use a series to solve limx → 0 [cos(x)/( − 4x)]
  • Use the MacLaurin Series of cos x to divide
limx → 0 [cos(x)/( − 4x)] = limx → 0 [(∑n = 0∞ [(( − 1)nx2n)/(2n)!] )/( − 4x)] = limx → 0 [(1 − [(x2)/2] + [(x4)/24] − [(x6)/720] + ...)/( − 4x)] = limx → 0 [1/( − 4x)] − [(x2)/( − 8x)] + [(x4)/(24( − 4x))] − [(x6)/(720( − 4x))] + ... = ∞− 0 + 0 − ... = ∞
Use a series to solve limx → 0 [(ex)/2x]
Use the MacLaurin Series of ex to divide
limx→0 [(ex)/2x] = limx → 0 [(∑n = 0∞ [(xn)/n!] )/2x] = limx→0 [(1 + x + [(x2)/2] + [(x3)/6] + ...)/2x] = limx→0 [1/2x] + [1/2] + [x/4] + [(x3)/12] + ... = ∞+ [1/2] + 0 + 0 + ... = ∞
Use a series to solve limx → 0 [(ln(x + 1))/x]
Use the MacLaurin Series of ln(x + 1) to divide
limx → 0 [(ln(x + 1))/x] = limx → 0 [(∑n = 0∞ [(( − 1)nxn)/n] )/x] = limx → 0 [(x − [(x2)/2] + [(x3)/3] − [(x4)/4] + ...)/x] = limx → 0 1 − [x/2] + [(x2)/3] + ... = 1 − + 0 − 0 = 1
Find the MacLaurin series of f(x) = e3x
Substitute (3x) for the MacLaurin series for exex = ∑n = 0∞ [(xn)/n!] e3x = ∑n = 0∞ [((3x)n)/n!] = 1 + 3x + [(9x2)/2] + [(9x3)/2] + [(27x4)/8] + ...
Find the first three times of the MacLaurin Series for the f(x) = − [6/((x + 1)4)]
  • Recognize − [6/((x + 1)4)] is the third derivative of [1/(x + 1)]
[(d3)/(dx3)]( [1/(x + 1)] ) = [(d3)/(dx3)]( 1 − x + x2 − x3 + x4 − x5 + ... ) = − 6 + 24x − 60x2 + ...

*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

Power Series Operations

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • 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

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John Zhu

John Zhu

Power Series Operations

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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