More on Sequences

More on Sequences - More on Sequences In the previous...

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More on Sequences In the previous section we introduced the concept of a sequence and talked about limits of sequences and the idea of convergence and divergence for a sequence. In this section we want to take a quick look at some ideas involving sequences. Let’s start off with some terminology and definitions. Given any sequence we have the following. 1. We call the sequence increasing if for every n . 2. We call the sequence decreasing if for every n . 3. If is an increasing sequence or is a decreasing sequence we call it monotonic . 4. If there exists a number m such that for every n we say the sequence is bounded below . The number m is sometimes called a lower bound for the sequence. 5. If there exists a number M such that for every n we say the sequence is bounded above . The number M is sometimes called an upper bound for the sequence. 6. If the sequence is both bounded below and bounded above we call the sequence bounded . Note that in order for a sequence to be increasing or decreasing it must be increasing/decreasing for every n . In other words, a sequence that increases for three terms and then decreases for the rest of the terms is NOT a decreasing sequence! Also note that a monotonic sequence must always increase or it must always decrease. Before moving on we should make a quick point about the bounds for a sequence that is bounded above and/or below. We’ll make the point about lower bounds, but we could just as easily make it about upper bounds. A sequence is bounded below if we can find any number m such that for every n . Note however that if we find one number m to use for a lower bound then any number smaller than m will also be a lower bound. Also, just because we find one lower bound that doesn’t mean there won’t be a “better” lower bound for the sequence than the one we found. In other words, there are an infinite number of lower bounds for a sequence that is bounded below, some will be better than others.
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In my class all that I’m after will be a lower bound. I don’t necessarily need the best lower bound, just a number that will be a lower bound for the sequence. Let’s take a look at a couple of examples. Example 1 Determine if the following sequences are monotonic and/or bounded. (a) [ Solution ] (b) [ Solution ] (c) [ Solution ] Solution (a) This sequence is a decreasing sequence (and hence monotonic) because, for every n . Also, since the sequence terms will be either zero or negative this sequence is bounded above. We
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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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More on Sequences - More on Sequences In the previous...

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