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Comparison & limit comparison test - Sequence and Series

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.

Lessons

Notes:
Note *The Comparison test says the following:
Let an\sum a_n and bn\sum b_n be two series where anbna_n\leq b_n for all nn and anbn0a_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 an0a_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.
  • 1.
    Overview:

  • 2.
    Convergence & Divergence of Comparison Tests
    Use the Comparison Test to determine if the series converge or diverge.
  • 3.
    Convergence & Divergence of Limit Comparison Tests
    Use the Limit Comparison Test to determine if the series converge or diverge.
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Comparison & limit comparison test

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