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Sequences - Sequences Lets start off this section with a...

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Sequences Let’s start off this section with a discussion of just what a sequence is. A sequence is nothing more than a list of numbers written in a specific order. The list may or may not have an infinite number of terms in them although we will be dealing exclusively with infinite sequences in this class. General sequence terms are denoted as follows, Because we will be dealing with infinite sequences each term in the sequence will be followed by another term as noted above. In the notation above we need to be very careful with the subscripts. The subscript of denotes the next term in the sequence and NOT one plus the n th term! In other words, so be very careful when writing subscripts to make sure that the “+1” doesn’t migrate out of the subscript! This is an easy mistake to make when you first start dealing with this kind of thing. There is a variety of ways of denoting a sequence. Each of the following are equivalent ways of denoting a sequence.
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In the second and third notations above a n is usually given by a formula. A couple of notes are now in order about these notations. First, note the difference between the second and third notations above. If the starting point is not important or is implied in some way by the problem it is often not written down as we did in the third notation. Next, we used a starting point of in the third notation only so we could write one down. There is absolutely no reason to believe that a sequence will start at . A sequence will start where ever it needs to start. Let’s take a look at a couple of sequences. Example 1 Write down the first few terms of each of the following sequences. (a) [ Solution ] (b) [ Solution ] (c) , where [ Solution ] Solution (a) To get the first few sequence terms here all we need to do is plug in values of n into the formula given and we’ll get the sequence terms.
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Note the inclusion of the “…” at the end! This is an important piece of notation as it is the only thing that tells us that the sequence continues on and doesn’t terminate at the last term. [ Return to Problems ] (b) This one is similar to the first one. The main difference is that this sequence doesn’t start at . Note that the terms in this sequence alternate in signs. Sequences of this kind are sometimes called alternating sequences. [ Return to Problems ] (c) , where This sequence is different from the first two in the sense that it doesn’t have a specific formula for each term. However, it does tell us what each term should be. Each term should be the n th digit of π. So we know that The sequence is then, [ Return to Problems ] In the first two parts of the previous example note that we were really treating the formulas as functions that can only have integers plugged into them. Or,
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This is an important idea in the study of sequences (and series). Treating the sequence terms as function evaluations will allow us to do many things with sequences that couldn’t do otherwise. Before delving further into this idea however we need to get a couple more ideas out of the way.
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