18. Sequences
Sometimes we are unable to compute a limit exactly. However, if we can at least show that the limit exists, then we can use some numerical technique to approximate the limit. We now introduce two properties of sequences (bounded and monotonic) which together help us show the limit exists.
c1. Bounded Sequences
Bounded Sequences
A sequence is
bounded above
if there is a number such that for all .
Then is called an upper bound.
A sequence is
bounded below
if there is a number such that for all .
Then is called an lower bound.
A sequence is
bounded if it is both bounded above and
bounded below.
Here are some examples:

Bounded above by .
Unbounded below.

Bounded above by .
Bounded below by .

Bounded below by .
Unbounded above.

Unbounded above.
Unbounded below.
Determine if each sequence is bounded above, below, neither or both. When appropriate, find upper and lower bounds.
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Solution
We study the sequence separately for odd and even.
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For odd, and which is always negative and becomes arbitrarily large and negative.
For even, and .
So the sequence is bounded above by (or by any positive number) but the sequence is not bounded below and so is not bounded. -
Solution
Since the terms are all positive, the sequence is bounded below by .
To see if there is an upper bound, we write the term as: Thus as long as , the term is the term times something bigger than , namely . So we suspect the sequence is unbounded above.
To see this we use a proof by contradiction. Suppose there is an upper bound . Then the term is which is times a number bigger than . So which contradicts the assumption that is an upper bound.
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