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Section 10.2 : More on Sequences

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

\[\left\{ {\frac{1}{{4n}}} \right\}_{n = 1}^\infty \]

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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.
Start Solution

For this problem let’s get the bounded information first as that seems to be pretty simple.

First note that because both the numerator and denominator are positive then the quotient is also positive and so we can see that the sequence must be bounded below by zero.

Next let’s note that because we are starting with \(n = 1\) the denominator will always be \(4n \ge 4 > 1\) and so we can also see that the sequence must be bounded above by one. Note that, in this case, this not the “best” upper bound for the sequence but the problem didn’t ask for that. For this sequence we’ll be able to get a better one once we have the increasing/decreasing information.

Because the sequence is bounded above and bounded below the sequence is also bounded.

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For the increasing/decreasing information we can see that, for our range of \(n \ge 1\), we have,

\[4n < 4\left( {n + 1} \right)\]

and so,

\[\frac{1}{{4n}} > \frac{1}{{4\left( {n + 1} \right)}}\]

If we define \({a_n} = \frac{1}{{4n}}\) this in turn tells us that \({a_n} > {a_{n + 1}}\) for all \(n \ge 1\) and so the sequence is decreasing and hence monotonic.

Note that because we have now determined that the sequence is decreasing we can see that the “best” upper bound would be the first term of the sequence or, \(\frac{1}{4}\).