Series1

Series1 - Series The Basics In this section we will...

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Series The Basics In this section we will introduce the topic that we will be discussing for the rest of this chapter. That topic is infinite series. So just what is a infinite series? Well, let’s start with a sequence (note the is for convenience, it can be anything) and define the following, The are called partial sums and notice that they will form a sequence, . Also recall that the is used to represent this summation and called a variety of names. The most common names are : series notation , summation notation , and sigma notation . You should have seen this notation, at least briefly, back when you saw the definition of a definite integral in Calculus I. If you need a quick refresher on summation notation see the review of summation notation in my Calculus I notes.

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Now back to series. We want to take a look at the limit of the sequence of partial sums, . Notationally we’ll define, We will call an infinite series and note that the series “starts” at because that is where our original sequence, , started. Had our original sequence started at 2 then our infinite series would also have started at 2. The infinite series will start at the same value that the sequence of terms (as opposed to the sequence of partial sums) starts. If the sequence of partial sums, , is convergent and its limit is finite then we also call the infinite series, convergent and if the sequence of partial sums is diverent then the infinite series is also called divergent . Note that sometimes it is convenient to write the infinite series as, We do have to be careful with this however. This implies that an infinite series is just an infinite sum of terms and as well see in the next section this is not really true. In the next section we’re going to be discussing in greater detail the value of an infinite series, provided it has one of course as well as the ideas of convergence and divergence. This section is going to be devoted mostly to notational issues as well as making sure we can do some basic manipulations with infinite series so we are ready for them when we need to be able to deal with them in later sections.
First, we should note that in most of this chapter we will refer to infinite series as simply series. If we ever need to work with both infinite and finite series we’ll be more careful with terminology, but in most sections we’ll be dealing exclusively with infinite series and so we’ll just call them series. Now, in the i is called the index of summation or just index for short and note that the letter we use to represent the index does not matter. So for example the following series are all the same. The only difference is the letter we’ve used for the index. It is important to again note that the index will start at whatever value the sequence of

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Series1 - Series The Basics In this section we will...

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