Series2 - Series Convergence/Divergence In the previous...

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Series Convergence/Divergence In the previous section we spent some time getting familiar with series and we briefly defined convergence and divergence. Before worrying about convergence and divergence of a series we wanted to make sure that we’ve started to get comfortable with the notation involved in series and some of the various manipulations of series that we will, on occasion, need to be able to do. As noted in the previous section most of what we were doing there won’t be done much in this chapter. So, it is now time to start talking about the convergence and divergence of a series as this will be a topic that we’ll be dealing with to one extent of another in almost all of the remaining sections of this chapter. So, let’s recap just what an infinite series is and what it means for a series to be convergent or divergent. We’ll start with a sequence and again note that we’re starting the sequence at only for the sake of convenience and it can, in fact, be anything. Next we define the partial sums of the series as,
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and these form a new sequence, . An infinite series, or just series here since almost every series that we'll be looking at will be an infinite series, is then the limit of the partial sums. Or, If the sequence of partial sums is a convergent sequence ( i.e. its limit exists and is finite) then the series is also called convergent and in this case if then, . Likewise, if the sequence of partial sums is a divergent sequence ( i.e. its limit doesn’t exist or is plus or minus infinity) then the series is also called divergent . Let’s take a look at some series and see if we can determine if they are convergent or divergent and see if we can determine the value of any convergent series we find. Example 1 Determine if the following series is convergent or divergent. If it converges determine its value. Solution To determine if the series is convergent we first need to get our hands on a formula for the general term in the sequence of partial sums. This is a known series and its value can be shown to be,
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Don’t worry if you didn’t know this formula (I’d be surprised if anyone knew it…) as you won’t be required to know it in my course. So, to determine if the series is convergent we will first need to see if the sequence of partial sums, is convergent or divergent. That’s not terribly difficult in this case. The limit of the sequence terms is, Therefore, the sequence of partial sums diverges to and so the series also diverges. So, as we saw in this example we had to know a fairly obscure formula in order to determine the convergence of this series. In general finding a formula for the general term in the sequence of partial sums is a very difficult process. In fact after the next section we’ll not be doing much with the partial sums of series due to the extreme difficulty faced in finding the general formula. This also means that we’ll not be doing much work with the value of series since in order to get the value we’ll also
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Series2 - Series Convergence/Divergence In the previous...

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