# College Calculus: Level II Alternating Series

Section 4: Sequences and Series: Lecture 5 | 25:26 min

In this lecture we are going to take a look at Alternating Series. The main test that we are going to be using is called the Alternating Series Test. In some Calculus classes, this is called the Leibnitz Alternating Series Test. We will be referring to it as AST for short. Sometimes you might see it as LAST for short. First we will see how this test works and which are the two conditions that must be satisfied to say that the series converges by the AST. Then we will talk about a very common mistake that Calculus students make when using AST. The second thing we are going to do with the AST is estimates of sums.

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 Mar 27, 2014 6:42 PM

Post by Brandyn Albrecht on March 27, 2014

Why can we take the (-1)^n out when solving for bn? Wouldn't the sign of the answer change as n goes from an even to odd number on it's way to infinity? I would think it would be like the sine function where it switches from positive to negative depending on the n being used. I could see taking a (+1)^n out but a negative I feel like wouldn't work because it's not always positive. Similar to Example 2 not always being positive.

3 answers

Last reply by: Dr. William Murray

Mon Nov 5, 2012 10:03 PM

Post by Monica Khun on August 8, 2012

why can't this be watched?