mes

Published on Jun 19, 2020
About :

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:

  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.

Download Video Notes: https://1drv.ms/b/s!As32ynv0LoaIh_NXBMhbCTS1dvztYg?e=EZPQ9u

View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-absolute-convergence-and-the-ratio-root-tests

Related Videos:

Sequences and Series Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz

Infinite Sequences: Limits, Squeeze Theorem, Fibonacci Sequence & Golden Ratio + MORE: https://peakd.com/mathematics/@mes/infinite-sequences-limits-squeeze-theorem-fibonacci-sequence-and-golden-ratio-more

Infinite Series: Definition, Examples, Geometric Series, Harmonics Series, Telescoping Sum + MORE: https://peakd.com/mathematics/@mes/infinite-series-definition-examples-geometric-series-harmonics-series-telescoping-sum-more

Infinite Sequences and Series: The Integral Test and Estimate of Sums: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-the-integral-test-and-estimate-of-sums

Infinite Sequences and Series: The Comparison Tests: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-the-comparison-tests

Infinite Sequences and Series: Alternating Tests: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-alternating-tests .


SUBSCRIBE via EMAIL: https://mes.fm/subscribe

DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate

Like, Subscribe, Favorite, and Comment Below!

Follow us on:

Official Website: https://MES.fm
Hive: https://peakd.com/@mes
Gab: https://gab.ai/matheasysolutions
Minds: https://minds.com/matheasysolutions
Twitter: https://twitter.com/MathEasySolns
Facebook: https://fb.com/MathEasySolutions
LinkedIn: https://mes.fm/linkedin
Pinterest: https://pinterest.com/MathEasySolns
Instagram: https://instagram.com/MathEasySolutions
Email me: contact@mes.fm

Try our Free Calculators: https://mes.fm/calculators

BMI Calculator: https://bmicalculator.mes.fm
Grade Calculator: https://gradecalculator.mes.fm
Mortgage Calculator: https://mortgagecalculator.mes.fm
Percentage Calculator: https://percentagecalculator.mes.fm

Try our Free Online Tools: https://mes.fm/tools

iPhone and Android Apps: https://mes.fm/mobile-apps

Tags :

mathematics messcience calculus science education

Comments:
Time until mes can give away $0 to their commenters.
0 Days 0 Hours 0 Minutes 0 Seconds
Woo! This creator can upvote comments using 3speak's stake today! Leave a quality comment relating to their content and you could receive an upvote worth at least a dollar.

Their limit for today is $0!
Reply:

To comment on this video please connect a HIVE account to your profile: Connect HIVE Account


More Videos
14:43
5 views 6 hours ago $
04:55
12 views 2 months ago $
04:00
4 views 2 months ago $
04:17
3 views a month ago $
05:57
0 views 2 weeks ago $
03:23
3 views 2 months ago $
00:24
32 views 2 months ago $