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- Sequence and Series
Comparison & limit comparison test
- Intro Lesson: a5:34
- Intro Lesson: b2:20
- Lesson: 1a14:04
- Lesson: 1b17:02
- Lesson: 1c11:51
- Lesson: 2a19:18
- Lesson: 2b8:06
- Lesson: 327:35
Comparison & limit comparison test
You may notice that some series look very complicated, but it shares the same properties as another series that looks very simple and easy. In this case, we can use the comparison test or limit comparison test. We will look at what conditions must be met to use these tests, and then use the tests on some complicated looking series. Lastly, we will use both the comparison test and the limit comparison test on a series, and conclude that they give the same result.
Basic Concepts: Introduction to infinite series, Convergence & divergence of geometric series , Divergence of harmonic series, P Series
Lessons
Note *The Comparison test says the following:
Let ∑an and ∑bn be two series where an≤bn for all n and anbn≥0. Then we say that
1. If ∑bn is convergent, then ∑an is also convergent
2. If ∑an is divergent, then ∑bn is also divergent.
The Limit Comparison Test says the following:
Let ∑an and ∑bn be two series where an≥0 and bn > 0 for all n. Then we say that
limn →∞ bnan=c
If c is a positive finite number, then either both series converge or diverge.
Let ∑an and ∑bn be two series where an≤bn for all n and anbn≥0. Then we say that
1. If ∑bn is convergent, then ∑an is also convergent
2. If ∑an is divergent, then ∑bn is also divergent.
The Limit Comparison Test says the following:
Let ∑an and ∑bn be two series where an≥0 and bn > 0 for all n. Then we say that
limn →∞ bnan=c
If c is a positive finite number, then either both series converge or diverge.
- IntroductionOverview:a)Comparison testb)Limit Comparison test
- 1.Convergence & Divergence of Comparison Tests
Use the Comparison Test to determine if the series converge or diverge.a)∑n=1∞2n+51b)∑n=1∞n5−sin4(2n)n4+5c)∑n=1∞n6n4cos4(7n)−1 - 2.Convergence & Divergence of Limit Comparison Tests
Use the Limit Comparison Test to determine if the series converge or diverge.a)∑n=3∞n8+n4n2+n3b)∑n=1∞n2−7n−121 - 3.Understanding of Both Tests
Use both the comparison and limit comparison test for the series ∑k=1∞k3−2k2+5k3−1. What do both tests say?
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5.
Sequence and Series
5.1
Introduction to sequences
5.2
Monotonic and bounded sequences
5.3
Introduction to infinite series
5.4
Convergence and divergence of normal infinite series
5.5
Convergence & divergence of geometric series
5.6
Convergence & divergence of telescoping series
5.7
Divergence of harmonic series
5.8
P Series
5.9
Alternating series test
5.10
Divergence test
5.11
Comparison & limit comparison test
5.12
Integral test
5.13
Ratio test
5.14
Root test
5.15
Absolute & conditional convergence
5.16
Radius and interval of convergence with power series
5.17
Functions expressed as power series
5.18
Taylor series and Maclaurin series
5.19
Approximating functions with Taylor polynomials and error bounds
Don't just watch, practice makes perfect
Practice topics for Sequence and Series
5.1
Introduction to sequences
5.2
Monotonic and bounded sequences
5.4
Convergence and divergence of normal infinite series
5.5
Convergence & divergence of geometric series
5.6
Convergence & divergence of telescoping series
5.7
Divergence of harmonic series
5.8
P Series
5.9
Alternating series test
5.10
Divergence test
5.11
Comparison & limit comparison test
5.12
Integral test
5.13
Ratio test
5.14
Root test
5.15
Absolute & conditional convergence
5.16
Radius and interval of convergence with power series
5.17
Functions expressed as power series
5.18
Taylor series and Maclaurin series