- Homework for 8.6 hasn't been graded yet.
- Taylor Homework is due today. I'll collect it in a moment.
- Hopefully you'll have both homework sets by Tuesday.
- Problems from 8.6: #84 and 87
- Problem #31 from 9.4
- The Taylor Polynomials approximate functions:
But, in approximating, we make errors: let's talk error now. Remember the error terms in the integration schemes (which approximate areas)? As we discussed for the case of Simpson's rule, it's nice to give an estimate -- but even nicer when you can provide an error estimate, also. Well, turns out that we can in the case of Taylor polynomials, too.
- Examples:
- #23, p. 500
- How do the Taylor and Maclaurin polynomials compare?
- animation
- #35, p. 500 (sin(x) as well -- Maclaurin polynomial)
- #46, p. 501
- #23, p. 500
-
Here's our 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 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
- #43-63, p. 544
What would we get if we could add up all these (infinite!) terms? (147.4131591025766....)