Series3 - Series Special Series In this section we are...

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Series Special Series In this section we are going to take a brief look at three special series. Actually, special may not be the correct term. All three have been named which makes them special in some way, however the main reason that we’re going to look at two of them in this section is that they are the only types of series that we’ll be looking at for which we will be able to get actual values for the series. The third type is divergent and so won’t have a value to worry about. In general, determining the value of a series is very difficult and outside of these two kinds of series that we’ll look at in this section we will not be determining the value of series in this chapter. So, let’s get started. Geometric Series A geometric series is any series that can be written in the form, or, with an index shift the geometric series will often be written as, These are identical series and will have identical values, provided they converge of course. If we start with the first form it can be shown that the partial sums are, The series will converge provided the partial sums form a convergent sequence, so let’s take the limit of the partial sums.
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Now, from Theorem 3 from the Sequences section we know that the limit above will exist and be finite provided . However, note that we can’t let since this will give division by zero. Therefore, this will exist and be finite provided and in this case the limit is zero and so we get, Therefore, a geometric series will converge if , which is usually written , its value is,
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Note that in using this formula we’ll need to make sure that we are in the correct form. In other words, if the series starts at then the exponent on the r must be n . Likewise if the series starts at then the exponent on the r must be . Example 1 Determine if the following series converge or diverge. If they converge give the value of the series. (a) [ Solution ] (b) [ Solution ] Solution (a) This series doesn’t really look like a geometric series. However, notice that both parts of the series term are numbers raised to a power. This means that it can be put into the form of a geometric series. We will just need to decide which form is the correct form. Since the series starts at we will want the exponents on the numbers to be . It will be fairly easy to get this into the correct form. Let’s first rewrite things slightly. One of the n ’s in the exponent has a negative in front of it and that can’t be there in the geometric form. So, let’s first get rid of that.
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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.

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Series3 - Series Special Series In this section we are...

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