- A calc teacher once said about his calculus class that "When I
finish ... I want the students to feel like they have
invented calculus and that only some accident of birth kept
them from beating Newton to the punch." Is that how you're
feeling? If not, what's holding you back?
Yesterday we used several concepts that are critical to an understanding of calculus:
- discretization
- linearization
- limits
- differentials
- infinite sums (as integrals)
- Return of your sequence quizzes, and comments.
- You have a homework due tomorrow -- any questions?
- Your quiz opportunity tomorrow is over 11.2: series
-
We discretize bits of the surface, and then make a series of little bands; take the limit, and get the general formula:
Examples:
- #19, p. 574
- #25
- #28
-
Here's a first example of a sequence:
- 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 must be true about the sum of the terms? Can we add up an infinite number of terms? This is the subject of the next section -- series.)
- Other sequences might be defined in terms of a function. For
example,
Examples:
- #3, p. 724
- #6
Other sequences are defined by "recurrence" (e.g. the Factorials; the Fibonacci numbers)
Example:
- #12, p. 724
Here's an especially interesting historical limit:
- One piece of good news: the obvious rules for limits are working:
Examples:
- #26, p. 724
- #27
- #29
- 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:
- #23, p. 724
- #83, p. 726
- #92
What would we get if we could add up all these (infinite!) terms? (147.4131591025766....)