Infinite Sequences and Series: Absolute Convergence and the Ratio/Root Tests

In this video I explore further into Infinite Sequences and Series and this time consider series that have involve the absolute values of a sequence. This involves the concept of “absolute convergence” which I show that it implies convergence in general; although the reverse is not always the case. In this case, the very interesting concept of “conditional convergence” arises which allows for the convergence of a series simply by re-arranging the terms such that the positive and negative terms cancel out; which is only possible if a series is not absolutely convergent. I also go over two specific tests to determine absolute convergence and they are the ratio and root test which are as their names describe. Half of this video is spent on 4 exercises at the end of the video and are a great way to reinforce the concepts introduced in this video.

The topics and their timestamps in the video are listed below:

- @ 0:00 - Intro
1. @ 1:59 - Absolute Convergence
- @ 2:43 - Definition 1
- @ 3:46 - Example 1
- @ 5:53 - Example 2
2. Conditional Convergence
- @ 9:22 - Definition 2
- @ 9:54 - Theorem 1
- @ 14:34 - Example 3
3. @ 20:46 - The Ratio Test
- @ 44:48 - Example 4
- @ 50:37 - Note on Estimating Sums
- @ 51:21 - Example 5
- @ 58:51 - Note on the Ratio Test
4. @ 1:01:25 - The Root Test
- @ 1:05:15 - Example 6
5. @ 1:08:46 - Rearrangements
6. Exercises
- @ 1:24:52 - Exercise 1
- @ 1:47:01 - Exercise 2: Proof of the Root Test
- @ 2:12:30 - Exercise 3
- @ 2:28:40 - Exercise 4

Please note the following list of corrections which I realized during the making of the video:

- @ 21:50 – Forgot to include the other condition of part (ii) of the Root Test: lim_(n→∞) |a_n+1/a_n| = ∞. Note that in the video I corrected this @ 36:02 during the proof of part (ii).
- @ 2:42:42 - Should’ve written |s_9 – r| = |1 – 5| = 4 less than |a_9| and also should've shown that we can ensure this to be the case regardless of the value of a_9.

Related Videos:

------------------------------------------------------

[DennisMathgod]

Very informative and interesting. Thanks for the upload!

[mes]

In this video I go over the very interesting concepts of infinite series of absolute values, absolute convergence, conditional convergence, Ratio and Root Tests, and much more!