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# FrontPage

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# Welcome to the IE Bridge Course-Wiki

 Announcement Date Regulation on bad weather Note: the tutorial will NOT be cancelled under Typhoon Signal No. 3 16th Aug Lecture evaluations 3rd  Aug MAXIMA worksheet     and     Reference Solution 3rd  Aug WXMAXIMA introduction handbook 3rd  Aug Maxima sourceforge 3rd  Aug

Class Sessions

Class Time : 10:30 - 12:30 , 14:00 - 16:00

 Item Tentative Date 0 MAXIMA, Review of elementary concepts, *Calculus Useful Formulae Aug. 5, 2012 1 limit of sequences Aug. 5, 2012 2 limit of a function Aug. 11, 2012 Aug. 19, 2012 4 Product rule Chain rule and Higher derivatives Aug. 19, 2012 5 Maximum and Minimum of a function Aug. 26, 2012 6 Indefinite Integrals Sept. 1, 2012 7 Definite Integrals Summary

Tutorial Materials

Time Slot : Thursday 19:00 - 20:00 , Sunday 16:30 - 17:30

Venue : ERB 1009

 Materials Date 1st tutorial Aug. 16/19, 2012 2nd tutorial Aug. 23/26, 2012 3rd tutorial Aug. 30/Sept. 1 , 2012 (Note: tutorial on Sunday is changed to Saturday this week only) 4th tutorial Sept. 6/9, 2012

Instructor's and TAs' contact details

 Professor/Lecturer/Instructor: Name: Prof. Sidharth Jaggi Office Location: SHB Room 706 Telephone: 2609-4326 Email: jaggi@ie.cuhk.edu.hk Other information: Office Hours: by appointment    Calendar

Teaching Assistant/Tutor:

Name:

#### Sheng Cai                                    Chris TC Wong

Office Location:

SHB Room 803                             SHB Room 702

Email:

cs010@ie.cuhk.edu.hk                   wtc012@ie.cuhk.edu.hk

Tutorial Time and Venue:

TBA #### sidjaggi said

at 7:54 pm on Aug 4, 2010

Hello world! #### ben choi said

at 12:46 am on Aug 9, 2010

Hi SID! I am Ben.
I would like to clarify whether my concept is correct after the course, and the study in these few hours.

"Consider an infinite sequence of numbers a1, a2, . . . , an, . . .. In many real-world situations, it tends to approach a constant number. We discuss rigorously the idea of a sequence converging towards a point called the limit. A number A is the limit of the sequence if the following is true: for any positive ϵ > 0, there exists an integer M (usually depends on ϵ) such that for all n > M,
|a n- A|<ϵ "

here is my guess after reading these page :
for example: here is a sequence"a1,a2......,am,.....,an". when the sequence approach to infin, it converge to A
You give me a "ϵ",a very small number. The relationship between am,A and ϵ is |am - A|=ϵ
it means whatever any ϵ you give me, there will be a mth number in that sequence may satisfied the condition:
|am - A|=ϵ
therefore, we should prove that in any terms m in the sequence, there still be n terms such that am<an.
when the condition |an - A|<ϵ, then we can prove that the limit of this sequence is A.

Am i right? #### sidjaggi said

at 3:27 am on Aug 9, 2010

Thanks, Ben, for your remarks. And good intuition. Almost correct, but you need to refine it a bit. Here are some questions to think about.

1. Is it always true that in any convergent sequence there is an m such that |am - A|=ϵ ? why did the definition say |a n- A|<ϵ rather than |am - A|=ϵ ? can you give an example where the = sign does not hold, but the < sign is correct?

2. does a convergent sequence have to be monotonic? (what is a monotonic function? look it up -- http://en.wikipedia.org/wiki/Monotonic_function )
can you give an example of a convergent sequence that is not monotonic? (hint -- we studied one in class.)
(just to get some more practice, how about the other way round -- can you give an example of a monotonic sequence that is not convergent?)

3. after thinking about 1. and 2., think about this question. is it true that if a_n is a convergent sequence, and |a_N- A|<ϵ for some N, than |a_(N+1)- A| is also always less than ϵ? if so, why? if not, give an example.

4. having thought of 3., do you now see why we say "there exists an integer M (usually depends on ϵ) such that for all n > M, |a n- A|<ϵ "

Apologies if it's frustrating for you to have questions in response to your question, but I think that thinking is the best way to learn something -- of course with hints :). Also, if we gave you all the answers, where's the fun? :)

Think about it for a while. If you have answers, feel free to post. Or not -- your choice. If you still have questions, feel free to post again. Or ask the tutors. Or meet me.

And to the others, feel free to try to answer the questions above, in this comment box.

Lastly, it'd be nice if you uploaded a photo to your profile, so I could see who's asking :) #### Tony said

at 1:28 am on Aug 10, 2010

Hi Jaggi, I am Tony

I would like to clarify something about "limit of sequence"

According to its definition, it claims

n->∞, f(n) -> L ... f(n) is a sequence { a_n }

when

for all positive real number ε, ∃N >0, ∀ n > N, | f(n) - L | < ε

=====

For example,

Use the definition to verify lim(n->∞) (sin n)/n = 0 ?

| (sin n)/n - 0 | < ε

| 1/n | < ε

1/ ε < n [ because ∃N >0, ∀ n > N ]

therefore we let N = 1/ε [!!! Q1. Is it a valid way to set the value of N ? ]

Suppose it is okay

| sin (1/ε) / (1/ε) | < | ε | | sin (1/ε) | < | ε | = ε

therefore, now we can proof lim(n->∞) (sin n)/n = 0 ... #### sidjaggi said

at 7:08 am on Aug 10, 2010

Hi Tony,

In answer to Question 1, yes! That's exactly what we did in class!

To get some more practice, now try proving that lim(n->∞) n does not exist. #### Tony said

at 10:09 am on Aug 10, 2010

- FIRST we suppose lim(n->∞) n exists, denoted by L

that means

for all positive real number ε, ∃N >0, ∀ n > N, | f(n) - L | < ε

when n is large enough, the difference between f(n) and the limit is sufficiently small ...

===

Now we suppose f(n) = n

when n->∞ , f(n)->∞

f(n) is strictly increasing with n and the sequence is said to diverge to infinity ...

Strictly speaking, | f(n) - lim(n->∞) n | is undefined, right ?

the difference between f(n) and L becomes very large ... [ we first suppose L is a fixed and unique real number, and f(n)->∞ when n->∞ ]

and it contradicts the definition of "limit of sequence" [ ! when n is large enough, the difference between f(n) and the limit is sufficiently small ]

therefore we can say lim(n->∞) n does not exist. #### sidjaggi said

at 10:34 am on Aug 10, 2010

Good! so now we've seen the power of this technique both for proving the existence of a limit, and for proving the non-existence of such a limit.

Just to make sure our intuition is good, here are a couple more problems to think about.

1. Does the sequence (-1)^n converge (if so, what's the limit/prove) or diverge (if so, prove)?

2. How about the sequence \sum_{i=1}^n 2^{-n}, that is, the sequence whose n'th term is the sum 1/2 + 1/4 + 1/8 + ... + 1/(2^n)

Just mental exercises to play with :)