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Math 33B, Spring
2000
Infinite Series
Daily notes
- 6/14 References to where to find proofs (a)-(d) have been posted
as Handout V, along with an explanation of what is wanted for (d).
- 6/14 Solutions to the sample final have been posted as Handout U.
- 6/12 My office hours Wednesday have to be moved because of a conflict.
Let's say 11:15-12:00.
- 6/12 Solutions #8, p. 16: This is from the chain rule!
- 6/12 Erratum: Homework solutions #8, p. 10: a 0 = 1/2,
so the constant term a 0 /2 should be 1/4. (Thanks to P.W.)
- 6/9 I may need to change my office hour listed for next Wednesday a.m.;
watch this page for details.
- 6/9 Homework papers are back with the TA's.
- 6/8 Vrej says he'll be doing a final review (Q&A session) on Wednesday,
June 14, 7-9pm, in Young CS 24. All are welcome.
- 6/8 Remus' office hours for exam week are T 2-3, W 12-2. (My own
hours are on Handout S.)
- 6/7 Suggestion on homework problems involving integrals like
t 2 e i n t from -pi
to pi : On your homework paper, first find the values of
such integrals with increasing powers of t , using
integration by parts, and then use this list in problems. As a
check: The value of the one just mentioned is 4 pi/n 2
(-1) n .
- 6/5 Someone's notebook found by Vrej; I have it now.
- 6/5 The quiz this week will have only the possibility (a), to derive the
relationship between complex and real Fourier coefficients, and not
(b) to derive Bessel's inequality, because I didn't get today's lecture
10-M posted in time to be fair. However, (b) will definitely be a
possibility for the final, so anyone who has already been studying it
will have an advantage. I'd suggest others study it this week too,
since it provides a good review of many ingredients from this course.
- 6/2 To provide examples of complex Fourier series, two have been
added as pages 14, 15, 16 of the on-line notes 9-F. The first is the
same as the example in the text, p. 388. The second is the one referred
to in Assignment 10.
- 6/1 Advance notice: The final exam room will be Rolfe 1200 .
- 6/1 See the updated posted version of Handout Q , listing
Problems Q-1 through Q-4. Previously an outdated version had
been posted with only three problems. The paper version handed
out in class was correct. (Thanks to one of you for pointing
this out.)
- 5/26 In today's notes (8F), pages 5 and 8 have been edited to show
terms in the order cos x, sin x, cos 2x, sin 2x, etc., corresponding
to the sum_n (a_n cos nx + b_n sin nx). The other order wasn't wrong,
but it's best to stick to this order.
- 5/24 The previous suggestion (5/22) has been amplified.
- 5/22 Sec. 6.1, Problem 20(b): A suggestion is to evaluate at x = pi/6
and then subtract off 1/3 of the series you got from evaluating at x = pi/2 .
Now you have half of the answer. (Thanks to Vrej.)
- 5/18 Again, TA Vrej Zarikian will hold a review session, to which
everyone is invited. This will be Sunday, May 21, 1:00-3:00 in MS 4000A.
- 5/17 Today's notes (7W) have been posted. Notice that only pages 1-4
cover material relevant for the next midterm. (For after the midterm: The
end of the lecture has been edited to show the reasoning better.)
- 5/17 In Problem M-5(b), by "take" it doesn't mean to take away, but
rather just to consider $100, $99, 99% of $99, and so on.
- 5/17 From a while back: In Lecture 5W, p. 9, there is a minor
correction in the division example, which shows on the web page.
- 5/17 In Problem M-6, the power series should be the sum of c_n x^n .
- 5/17 In the notes for Lecture 6-F, p. 4, it should have said
|x1| = r|x2|, rather than the other way around. This has been fixed on
line. (This is important, since it's a quiz and midterm topic.) (Thanks
to EG.)
- 5/16 As we start Chapter 6: p. 358 is misprinted in the course
booklet. You should have an insert with the correction. It's also
here.
- 5/16 See an article on Fourier ,
an important figure during turbulent times.
- 5/12 Also, the due date of Assignment 7 should say Friday, May 19
(not 18). This has been fixed on-line.
- 5/12 In the paper version of Handout M (Assignment 7), an
erratum: Problem M-2: should say series resembles the series
for the hyperbolic cosine, cosh, since there is no
alternation of sign. (The correction has been made in the on-line version.)
- 5/10 Some demos have been added above. To run them you'll need to
have Java turned on in your browser. Within a week you'll be able
to understand the first two. (The third and fourth are not so relevant for
this course.)
- 5/10 For today's lecture (6-W), in the on-line notes I have
added a computer plot of a sequence of functions we discussed.
- 5/8 Although coming to lecture is optional, here are some interesting
statistics: The median grade of people who picked up their midterms in
class last Wednesday was B, while the median grade of people who didn't
pick them up then was C-.
- 5/8 Lecture 6-M has been posted. The last several pages have been
changed quite a bit to use a method that will help with the homework.
- 5/8 Midterm Problem 1(b): You may ask for a regrade if
you had a point taken off for simply saying the given series is
a constant times the standard series sum n
1/n!, which converges. The intention was that the problem
would be answered with the ratio test, but I agree the problem
as stated could legitimately be given this kind of answer.
- 5/8 Misprint on handout L, Problem L-3(f): x should be n.
(Thanks to Z.V. for the report.)
- 5/7 In Section 3.1, p. 97, Ex 36, the k is intended to be
an unspecified constant. For this problem, try a substitution of something
clever for x , as in lecture.
- 5/4 Notes for 5-W have now been posted. There is one clarification;
see VOH.
- 5/3 Some people have been asking about Problem J-1(b). Part (b)
is supposed to be the same as Part (a) except with 1/N instead
of 1/6.
- 4/28 A couple of things were added to today's lecture notes before posting.
- 4/27 On Monday, please bring your UCLA ID to the exam. Thanks.
- 4/27 See a new VOH posting for advice on harder problems in Section 3.9.
- 4/27 TA Vrej Zarikian will hold a review session on Sunday,
April 30, from 1:00 to 3:00 p.m. in MS 5200. Students
from all discussion sections are welcome.
- 4/27 Homework solutions to Assignments 1-3 are now available,
thanks to TA Vrej Z.--see the link above. Vrej cautions that he
originally wrote these for his own use, so they are somewhat abbreviated.
- 4/26 In today's posted notes (4-M), I did some editing in the
first and last parts to make it clearer.
- 4/26 As announced in class, Section 3.9 will not be on Midterm 1.
Also, you can postpone problems from this section until the following
homework assignment. But do the rest of Assignment 4 this week.
- 4/21 A simple question to consider: In this morning's lecture
we first discussed the Mean Value Theorem (MVT) and then the Taylor
remainder R_n in the derivative form. If instead you start by
knowing the Taylor remainder formula, which n gives you the MVT?
- 4/21 See a new VOH posting re. how to choose a and b in remainder
problems.
- 4/20 One of you has pointed out that on the posted lecture notes for
Monday (3M), p. 4 [starting with example 2] was out of order; it should have
been the last page. This has been fixed. If your browser still shows it
out of order, it may be using cached copies; in that case you'll need to
click on the "reload" button. (Thanks to D. G.)
- 4/19 In general and more specifically in p. 126, Ex. 29 and 30,
when the authors say "interval of convergence" it's not entirely clear
if they are referring to the full interval of convergence, which includes
one or both endpoints if the series converges there. In lecture, though,
I have always meant the full interval, so do look at the endpoints
in these problems. In 29a [not to hand in] the endpoints are hard.
For practice, try them anyway using this fact: the n-th term has a
value between 1/2n and 1/sqrt(n) . (The 1/2n part is explained
on VOH.) Your answer may be different from the one in the back of the book.
- 4/19: See VOH for an idea on p. 126, Ex. 30 (and 29a).
- 4/14: Today's lecture notes have been edited for clarity before
posting on the web. In lecture I had written a_n instead of a_N
a couple of times. Also, one of you suggested that in the
"giving up a little" method it might be better to use comparison
by division, and I agree, so the examples have been changed
accordingly.
- 4/13: If you are checking this page more than once during a
session on a PC, be sure to click on the "RELOAD" button of your
browser. Otherwise your browser may use an old cached copy and
you won't see any additions to the page.
- 4/13: There is a misprint in Problem 14 on p. 119, Section
3.4: The exponent of 2 should be n. (Thanks to the TA's and
K.L. for pointing this out.)
- 4/13: If you want to ask a question by email, if it might be of
interest to other people I suggest that you post it on Virtual Office
Hours (VOH), either with or without giving your name. But also please
send me email saying to check VOH, so I'll notice it sooner.
- 4/13: See Virtual Office Hours for several questions and responses.
- 4/8: I'm told that UCLA Store ran out of the reprinted book for
the course but that they are making more copies.
- 4/7: My Friday (1-F) lecture was rushed at the end, so I
rewrote the part about the integral test before posting the notes.
- 4/5: For Friday's homework, you'll need to read about the
comparison test for series whose terms are >= 0. It's easy: How
do the terms of the series compare to a series you know about, such
as the harmonic series with terms 1/n (which diverges) or a geometric
series with r < 1 (which converges). A series with terms bigger than
terms of some divergent series must diverge; a series with terms
smaller than terms of some convergent series must converge.
- 4/5: In Problem C-7, not all three give something from B-(8)
through B-(16), but you should still get something interesting for
the one that doesn't.
- 4/5: The new section has now been added, 1e. If you were on a
Tuesday waitlist before you should have been put in it, so check and
then attend it.
- 4/5: Notice that the solutions to odd-numbered problems are in the
back of the book.
- 4/5: We are adding another discussion section, Tuesday at 9:00.
This means we'll be able to accommodate everyone currently on the wait
list, plus the couple of people from Extension. For those involved
who are not able to attend Tuesdays, we'll work around it. If you are
currently in a Thursday section and would like to switch to Tuesday,
check with me.
- 4/3: Erratum: In Handout B, (24) should say |x| <= 1. This has
been corrected in the on-line version.
- 4/3: Vrej's office hours are Tuesday 12-1 and Thursday 10-11.
Remus' office hours are Thursday 10:30-11:30 and 2:00 to 3:00.
