Ratio Tes1

Ratio Tes1 - Ratio Test In this section we are going to...

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Ratio Test In this section we are going to take a look at a test that we can use to see if a series is absolutely convergent or not. Recall that if a series is absolutely convergent then we will also know that it’s convergent and so we will often use it to simply determine the convergence of a series. Before proceeding with the test let’s do a quick reminder of factorials. This test will be particularly useful for series that contain factorials (and we will see some in the applications) so let’s make sure we can deal with them before we run into them in an example. If n is an integer such that then n factorial is defined as, Let’s compute a couple real quick.
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In the last computation above, notice that we could rewrite the factorial in a couple of different ways. For instance, In general we can always “strip out” terms from a factorial as follows. We will need to do this on occasion so don’t forget about it.
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Also, when dealing with factorials we need to be very careful with parenthesis. For instance, as we can see if we write each of the following factorials out. Again, we will run across factorials with parenthesis so don’t drop them. This is often one of the more common mistakes that students make when the first run across factorials. Okay, we are now ready for the test. Ratio Test Suppose we have the series . Define, Then, 1. if the series is absolutely convergent (and hence convergent). 2. if the series is divergent. 3. if the series may be divergent, conditionally convergent, or absolutely convergent. A proof of this test is at the end of the section. Notice that in the case of the ratio test is pretty much worthless and we would need to resort to a different test to determine the convergence of the series.
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Also, the absolute value bars in the definition of L are absolutely required. If they are not there it will be possible for us to get the incorrect answer. Let’s take a look at some examples.
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Ratio Tes1 - Ratio Test In this section we are going to...

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