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### Section 10.2 : More on Sequences

2. Determine if the following sequence is increasing, decreasing, not monotonic, bounded below, bounded above and/or bounded.

$\left\{ {n{{\left( { - 1} \right)}^{n + 2}}} \right\}_{n = 0}^\infty$

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Hint : There is no one set process for finding all this information. Sometimes it is easier to find one set of information before the other and at other times it doesn’t matter which set of information you find first. This is one of those sequences that it doesn’t matter which set of information you find first and both sets should be fairly easy to determine the answers without a lot of work.
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For this problem let’s get the increasing/decreasing information first as that seems to be pretty simple and will help at least a little bit with the bounded information.

In this case let’s just write out the first few terms of the sequence.

$\left\{ {n{{\left( { - 1} \right)}^{n + 2}}} \right\}_{n = 0}^\infty = \left\{ {0,\,\, - 1,\,\,2,\,\, - 3,\,\,4,\,\, - 5,\,\,6,\,\, - 7, \ldots } \right\}$

Just from the first three terms we can see that this sequence is not an increasing sequence and it is not a decreasing sequence and therefore is not monotonic.

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Now let’s see what bounded information we can get.

From the first few terms of the sequence we listed out above we can see that each successive term will get larger and change signs. Therefore, there cannot be an upper or a lower bound for the sequence. No matter what value we would try to use for an upper or a lower bound all we would need to do is take $$n$$ large enough and we would eventually get a sequence term that would go past the proposed bound.

Therefore, this sequence is not bounded.