Here's our first example of a sequence:
What would we get if we could add up all these (infinite!) terms?
(147.4131591025766....)
- Definition: A sequence is a function f(n) whose
domain is a subset of the integers. The values
})
are called the terms of the sequence
and n is called the index. For most of our sequences, the
domain will be the natural numbers: {1, 2, 3, ....}.
- In the example sequence above, what do you suppose is the limit of
the sequence as
?
- There are lots of other sequences that we might encounter
naturally: for example, there's one in probability and statistics that
is closely related to the one we just looked at: the Poisson
distribution, which has as its probabilities the sequence
(what do you notice about the sum of the terms?).
- Other sequences might be defined in terms of a function. For
example,
},\hspace{.1in}{\mbox{where}}\hspace{.1in}f(x)=\frac{1}{x})
Examples:
Other sequences are defined by "recurrence" (e.g. the Fibonacci numbers)
Example:
Here's an especially interesting historical limit:
- One piece of good news: the obvious rules for limits are working:
Examples:
- Some vocabulary: bounded sequences....
And some theorems related to this notion:
(especially useful for alternating sequences)
(note that the converse is false)
Let's use Theorem 6 to show that the sequence we initially considered is
convergent:
- Geometric sequence: a sequence of the form
where r is called the common ratio.
- Examples:
-- how did I know?
