# College Calculus: Level II Series

Section 4: Sequences and Series: Lecture 2 | 31:46 min

In this tutorial we are going to talk about Series. Before we look at some examples we are going to talk about several bits of notations and definitions. So, first, we are going to talk about Sigma notation and what does it stands for. Then we are going to talk about Sequence of Partial Sums. Later on, we are going to talk about Geometric Series and we will introduce its formula. One more theorem that we are going to talk about is Test for Divergence. At the end we are going to do several examples.

Table of Contents

24m 52s

- Intro0:00
- Important Equation0:07
- Where It Comes From (Product Rule)0:20
- Why Use It?0:35
- Lecture Example 11:24
- Lecture Example 23:30
- Shortcut: Tabular Integration7:34
- Example7:52
- Lecture Example 310:00
- Mnemonic: LIATE14:44
- Ln, Inverse, Algebra, Trigonometry, e15:38
- Additional Example 4-1
- Additional Example 5-2

25m 30s

- Intro0:00
- Important Equation0:07
- Powers (Odd and Even)0:19
- What To Do1:03
- Lecture Example 11:37
- Lecture Example 23:12
- Half-Angle Formulas6:16
- Both Powers Even6:31
- Lecture Example 37:06
- Lecture Example 410:59
- Additional Example 5-1
- Additional Example 6-2

30m 9s

- Intro0:00
- Important Equations0:06
- How They Work0:35
- Example1:45
- Remember: du and dx2:50
- Lecture Example 13:43
- Lecture Example 210:01
- Lecture Example 312:04
- Additional Example 4-1
- Additional Example 5-2

41m 22s

- Intro0:00
- Overview0:07
- Why Use It?0:18
- Lecture Example 11:21
- Lecture Example 26:52
- Lecture Example 313:28
- Additional Example 4-1
- Additional Example 5-2

20m

- Intro0:00
- Using Tables0:09
- Match Exactly0:32
- Lecture Example 11:16
- Lecture Example 25:28
- Lecture Example 38:51
- Additional Example 4-1
- Additional Example 5-2

22m 36s

- Intro0:00
- Trapezoidal Rule0:13
- Graphical Representation0:20
- How They Work1:08
- Formula1:47
- Why a Trapezoid?2:53
- Lecture Example 15:10
- Midpoint Rule8:23
- Why Midpoints?8:56
- Formula9:37
- Lecture Example 211:22
- Left/Right Endpoint Rule13:54
- Left Endpoint14:08
- Right Endpoint14:39
- Lecture Example 315:32
- Additional Example 4-1
- Additional Example 5-2

21m 8s

- Intro0:00
- Important Equation0:03
- Estimating Area0:28
- Difference from Previous Methods0:50
- General Principle1:09
- Lecture Example 13:49
- Lecture Example 26:32
- Lecture Example 39:07
- Additional Example 4-1
- Additional Example 5-2

44m 18s

- Intro0:00
- Horizontal and Vertical Asymptotes0:04
- Example: Horizontal0:16
- Formal Notation0:37
- Example: Vertical1:58
- Formal Notation2:29
- Lecture Example 15:01
- Lecture Example 27:41
- Lecture Example 311:32
- Lecture Example 415:49
- Formulas to Remember18:26
- Improper Integrals18:36
- Lecture Example 521:34
- Lecture Example 6 (Hidden Discontinuities)26:51
- Additional Example 7-1
- Additional Example 8-2

23m 20s

- Intro0:00
- Important Equation0:04
- Why It Works0:49
- Common Mistake1:21
- Lecture Example 12:14
- Lecture Example 26:26
- Lecture Example 310:49
- Additional Example 4-1
- Additional Example 5-2

28m 53s

- Intro0:00
- Important Equation0:05
- Surface Area0:38
- Relation to Arclength1:11
- Lecture Example 11:46
- Lecture Example 24:29
- Lecture Example 39:34
- Additional Example 4-1
- Additional Example 5-2

24m 37s

- Intro0:00
- Important Equation0:09
- Main Idea0:12
- Different Forces0:45
- Weight Density Constant1:10
- Variables (Depth and Width)2:21
- Lecture Example 13:28
- Additional Example 2-1
- Additional Example 3-2

25m 39s

- Intro0:00
- Important Equation0:07
- Main Idea0:25
- Centroid1:00
- Area1:28
- Lecture Example 11:44
- Lecture Example 26:13
- Lecture Example 310:04
- Additional Example 4-1
- Additional Example 5-2

22m 26s

- Intro0:00
- Important Equations0:05
- Slope of Tangent Line0:30
- Arc length1:03
- Lecture Example 11:40
- Lecture Example 24:23
- Lecture Example 38:38
- Additional Example 4-1
- Additional Example 5-2

30m 59s

- Intro0:00
- Important Equations0:05
- Polar Coordinates in Calculus0:42
- Area0:58
- Arc length1:41
- Lecture Example 12:14
- Lecture Example 24:12
- Lecture Example 310:06
- Additional Example 4-1
- Additional Example 5-2

31m 13s

- Intro0:00
- Definition and Theorem0:05
- Monotonically Increasing0:25
- Monotonically Decreasing0:40
- Monotonic0:48
- Bounded1:00
- Theorem1:11
- Lecture Example 11:31
- Lecture Example 211:06
- Lecture Example 314:03
- Additional Example 4-1
- Additional Example 5-2

31m 46s

- Intro0:00
- Important Definitions0:05
- Sigma Notation0:13
- Sequence of Partial Sums0:30
- Converging to a Limit1:49
- Diverging to Infinite2:20
- Geometric Series2:40
- Common Ratio2:47
- Sum of a Geometric Series3:09
- Test for Divergence5:11
- Not for Convergence6:06
- Lecture Example 18:32
- Lecture Example 210:25
- Lecture Example 316:26
- Additional Example 4-1
- Additional Example 5-2

23m 26s

- Intro0:00
- Important Theorem and Definition0:05
- Three Conditions0:25
- Converging and Diverging0:51
- P-Series1:11
- Lecture Example 12:19
- Lecture Example 25:08
- Lecture Example 36:38
- Additional Example 4-1
- Additional Example 5-2

22m 44s

- Intro0:00
- Important Tests0:01
- Comparison Test0:22
- Limit Comparison Test1:05
- Lecture Example 11:44
- Lecture Example 23:52
- Lecture Example 36:01
- Lecture Example 410:04
- Additional Example 5-1
- Additional Example 6-2

25m 26s

- Intro0:00
- Main Theorems0:05
- Alternation Series Test (Leibniz)0:11
- How It Works0:26
- Two Conditions0:46
- Never Use for Divergence1:12
- Estimates of Sums1:50
- Lecture Example 13:19
- Lecture Example 24:46
- Lecture Example 36:28
- Additional Example 4-1
- Additional Example 5-2

33m 27s

- Intro0:00
- Theorems and Definitions0:06
- Two Common Questions0:17
- Absolutely Convergent0:45
- Conditionally Convergent1:18
- Divergent1:51
- Missing Case2:02
- Ratio Test3:07
- Root Test4:45
- Lecture Example 15:46
- Lecture Example 29:23
- Lecture Example 313:13
- Additional Example 4-1
- Additional Example 5-2

38m 36s

- Intro0:00
- Main Definitions and Pattern0:07
- What Is The Point0:22
- Radius of Convergence Pattern0:45
- Interval of Convergence2:42
- Lecture Example 13:24
- Lecture Example 210:55
- Lecture Example 314:44
- Additional Example 4-1
- Additional Example 5-2

30m 18s

- Intro0:00
- Taylor and Maclaurin Series0:08
- Taylor Series0:12
- Maclaurin Series0:59
- Taylor Polynomial1:20
- Lecture Example 12:35
- Lecture Example 26:51
- Lecture Example 311:38
- Lecture Example 417:29
- Additional Example 5-1
- Additional Example 6-2

50m 50s

- Intro0:00
- Main Formulas0:06
- Alternating Series Error Bound0:28
- Taylor's Remainder Theorem1:18
- Lecture Example 13:09
- Lecture Example 29:08
- Lecture Example 317:35
- Additional Example 4-1
- Additional Example 5-2

For more information, please see full course syllabus of College Calculus: Level II

1 answer

Last reply by: Dr. William Murray

Thu Apr 24, 2014 6:13 PM

Post by Jack Miars on April 19, 2014

In lecture example three, when solving for s_n you put that the terms that don't get cancelled are 1/n+1 and 1/n+2, however if you look at it for when n=3, the values are 5 (n+2) and 6 (n+3). Is this an error or am I mistaken?

3 answers

Last reply by: Dr. William Murray

Wed Nov 20, 2013 1:41 PM

Post by Narin Gopaul on October 22, 2013

Good day DR. I just took my second test in calculus and failed miserably. I used your videos which did help me to get the theory down. The problem am facing is that the (Functions) you see on a test where you will apply the theory on to solve is extremely hard. How do I solve this problem when there is no telling of what type of functions could appear.

I will appreciate any advice you can give!!!

1 answer

Last reply by: Dr. William Murray

Sun Sep 22, 2013 8:38 AM

Post by Mark Bogenrieder on September 17, 2013

Looking at this particular problem, what exactly clues you in to use partial fraction decomposition? You go right to PFD without really explaining what led you to that solution. Thanks.

1 answer

Last reply by: Dr. William Murray

Thu Apr 25, 2013 2:03 PM

Post by vishal patel on April 24, 2012

Wish I had Professor Murray for my course.

1 answer

Last reply by: Dr. William Murray

Thu Apr 25, 2013 2:02 PM

Post by Luis Robles on March 27, 2012

Great explanation!

Keep up the good work!