Infinite Sequences and Series: The Comparison Tests

in hive-181335 •  16 days ago 

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In this video I go further into the wonderful world of infinite sequences and series and this time look at using comparisons with known series to test if a given series is convergent or divergent. The standard Comparison Test simply shows that if the terms of a series are less than the terms of a known convergent series, such as a p-series or geometric series, than the given series is also convergent; and vice versa if the terms are greater than the terms of a divergent series so too is the given series divergent. I go over many examples and exercises illustrating this test as well as go over a similar Limit Comparison Test. This particular test involves taking the limit as the number of terms approach infinity of the ratio of a given series to a known series; if the ratio approaches a number greater than 0 than both series are either both divergent or both convergent. In later exercises I show that if the ratio approaches 0 and the known series is convergent then so too is the given series; but if the ratio approaches infinity and the known series is divergent than so too is the given series. Many more examples and details are covered in this video so hope you enjoy!

The topics and sections covered in this video are listed below with their timestamps:

  1. @ 1:25 - Introduction to the Comparison Tests
  2. @ 10:27 - The Comparison Test
    • @ 25:10 - Example 1
    • @ 29:18 - Note #1
    • @ 29:55 - Example 2
    • @ 36:24 - Note #2
  3. @ 38:12 - The Limit Comparison Test
    • @ 42:35 - Example 3
    • @ 45:34 - Example 4
  4. @ 55:52 - Estimating Sums
    • @ 1:00:19 - Example 5
  5. @ 1:10:31 - Exercises
    • @ 1:10:51 - Exercise 1
    • @ 1:28:16 - Exercise 2
    • @ 1:41:13 - Exercise 3
    • @ 1:50:52 - Exercise 4

Download Video Notes:

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

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

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


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In this video I go over another in-depth tutorial on infinite sequences and series and this time show that we can compare the convergence or divergence of a known series as a test to see if a given series converges or diverges accordingly.

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