Comparison & limit comparison test

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\sum a_n and ∑bn\sum b_n be two series where an≤bna_n\leq b_n for all nn and anbn≥0a_nb_n\geq0. Then we say that
1. If ∑bn\sum b_n is convergent, then ∑an\sum a_n is also convergent
2. If ∑an\sum a_n is divergent, then ∑bn\sum b_n is also divergent.

The Limit Comparison Test says the following:
Let ∑an\sum a_n and ∑bn\sum b_n be two series where an≥0a_n\geq 0 and bnb_n > 0 for all nn. Then we say that

lim⁡\limn →∞\infty anbn=c\frac{a_n}{b_n}=c

If cc is a positive finite number, then either both series converge or diverge.
  • Introduction
    Overview:

    a)
    Comparison test

    b)
    Limit Comparison test

  • 1.
    Convergence & Divergence of Comparison Tests
    Use the Comparison Test to determine if the series converge or diverge.
    a)
    ∑n=1∞12n+5 \sum_{n=1}^{\infty}\frac{1}{2^n+5}

    b)
    ∑n=1∞n4+5n5−sin4(2n) \sum_{n=1}^{\infty}\frac{n^4+5}{n^5-sin^4(2n)}

    c)
    ∑n=1∞n4cos4(7n)−1n6 \sum_{n=1}^{\infty}\frac{n^4cos^4(7n)-1}{n^6}

  • 2.
    Convergence & Divergence of Limit Comparison Tests
    Use the Limit Comparison Test to determine if the series converge or diverge.
    a)
    ∑n=3∞n2+n3n8+n4 \sum_{n=3}^{\infty}\frac{n^2+n^3}{\sqrt{n^8+n^4}}

    b)
    ∑n=1∞1n2−7n−12 \sum_{n=1}^{\infty}\frac{1}{n^2-7n-12}

  • 3.
    Understanding of Both Tests
    Use both the comparison and limit comparison test for the series ∑k=1∞k3−1k3−2k2+5\sum_{k=1}^{\infty}\frac{\sqrt{k^3-1}}{k^3-2k^2+5} . 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
Calculus 2
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