- Quiz on Sequences returned
- Yikes! You guys aren't doing the reading.
- Homework for 8.6 returned. Comments from the grader:
- #2: Several computed
as
!
- #4-44: Many invoked Theorem 1 to check for convergence, but then failed to compare it to the actual function (no comparison tests)
- #52, 62: some didn't use absolute value correctly
- #79: most did okay
- #84: Most did not understand what to do.
- #2: Several computed
- Taylor polynomial homework will be returned Thursday.
- Homework on sequences is due on Thursday
- Read Section 11.2 for next time, responding to the worksheet for section 11.2.
- 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,
Examples:
- #4, p. 543
- #11, p. 543
Here's an especially interesting historical limit:
- One piece of good news: the obvious rules for limits are working:
Examples:
- #12, p. 543
- #43, p. 544
- 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:
- #19, p. 543
- #28, p. 544
- #45, p. 544
- #47, p. 544. Hint:
- The sequence of the Fibonacci numbers and some Fibonacci fun...
