Review Question 5: Recap on Tests for Convergence or Divergence of Infinite Series

Math Easy Solutions

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In this video I recap on 7 different types of tests for determining if an infinite series converges or diverges. The first is the Test for Divergence which states that if the limit of the sequence of terms does not approach 0, then the series is divergent. The second is the Integral Test, which states that the series is only convergent if its corresponding integral is also convergent. The third is the Comparison Test, which states that a series is convergent if every term of its sequence is smaller than that of a known convergent series. The fourth is the Limit Comparison Test, which states that if the limit of the ratio of the terms of 2 sequences is equal to a finite number c that is greater than 0, then either both series converge or both diverge. The fifth is the Alternating Series Test, which states that if the absolute value of all the terms of an alternating series is less than each subsequent term, and their limit approaches 0, then the alternating series is convergent. The sixth is the Ratio Test, which states that if the limit of the absolute value of the ratio of the n + 1 term divided by the n-th term is equal to a finite number less than 1, then the series is convergent. The seventh test is the Root Test, which states that if the n-th root of the absolute value of the n-th term is equal to a finite number less than 1, then the series is convergent.

Note that for all the tests listed, they have similar conditions for when the series is divergent, or in the case of the Ratio and Root Tests, where the tests are inconclusive when the limits equal to 1.

The timestamps of key parts of the video are listed below:

Question 5: 0:00
(a) Test for Divergence: 0:22
(b) Integral Test: 1:25
(c) Comparison Test: 3:32
(d) Limit Comparison Test: 4:43
(e) Alternating Series Test: 5:28
(f) Ratio Test: 8:01
(h) Root Test: 9:23

This video was taken from my earlier video listed below:

Infinite Sequences and Series: Review and True-False Quiz: https://youtu.be/F0dsQLdXXpI
HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-review-and-true-false-quiz
Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FCqXVJv1r7eJvrvphfkr6L