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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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View Full Document 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.
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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View Full Document 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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This note was uploaded on 11/10/2011 for the course MATH 136 taught by Professor Prellis during the Fall '08 term at Rutgers.
 Fall '08
 prellis
 Factorials

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