1. Ch 5.1 Fundamental Concepts of Infinite Series
講者： 許永昌 老師

2. Road-map of Chapter 5 Sequence (序列) Series (級數) Bounded
Convergence and divergence
Series (級數)
Cauchy criterion
Convergent test
lim un=0.
For positive un
Comparison Test
Cauchy root Test
Ratio Test
Integral Test
Absolute and Conditional Convergence
Series of Functions
Uniform convergence
Weierstrass M test
Abel’s test: (i)un(x)=anfn(x), (ii)0  fn+1(x)fn(x)M and (iii)the infinite series of {an} is convergent.
Taylor’s expansion and power series
Asymptotic Series

3. Contents Definition of sequence and infinite series
An example of sequence: damped oscillation
The concept of bounded and convergent
Convergence and Cauchy sequence.
Addition and Multiplication of series

4. Definition of Sequence and Series
A sequence of numbers: an ordered set of numbers in one-to-one correspondence with the positive integers; write {un}{u1, u2, … }.
A series: The sum of terms of a sequence.
Partial sums

5. An example of sequence: damped oscillation
t=1,2,3,… (s)
It is Bounded
It is Convergent
Q: 怎麼文字表達它們？

6. Bounded and convergent
Reference: Introduction to Mathematical Physics, M.T. Vaughn
The sequence {un} is bounded if there is some positive number M such that |un|<M for all positive integers n.
In this example, M=2.
The sequence {un} is convergent to the limit u if for every e >0 there is a positive number N such that |un-u|<e whenever n>N.
In this case, u=1.
N=250,e=0.2 for dotted lines.
un+1 un e u

7. Cauchy criterion (請預讀P260)
Cauchy sequence:
The sequence {un} is a Cauchy sequence if for every e >0 there is a positive integer N such that |up-uq|< e whenever p,q >N.
請與上一頁的說法比較一下，看看差別在哪。
Cauchy criterion:
A sequence is convergent if and only if it is a Cauchy sequence.

8. Example: A drunken man
P0(2,1): the probability for this man to go freely from 1 to 2.
P(A): the probability for this man to leave the “Alice bar”.
P(2,1): The probability for this drunken man to go back home.
Reference: A Guide to Feynman Diagrams in the Many-Body Problem.
P(2,1)=P0(2,1)+ P0(2,A)P(A) P0(A,1)+ P0(2,A)P(A) P0(A,A)P(A) P0(A,1)+… 2 2 A 2 2 = + A +
+ … A 1 1 1 1

9. Sequence and Series If we define P0(A,A)P(A)=r, we get
Which is an infinite series.
Sequence: {0.5n}
Partial sums
Converged? S=?

10. Necessary condition for the convergence of a series (請預讀P259)
The necessary condition is
Exercise: Prove it.
However, it is not sufficient to guarantee convergence.
In Ch5.2 it will tell us the general 4 kinds of test.
Theorem:
If the (i) un>0 are (ii) monotonic decreasing to zero, that is, un > un+1 for all n, then Snun is converging to S if, and only if, sn-nun converges to S.
Prove: Hint:
tnsn-nun<sn. sn sn tn

11. Example of divergent series: Harmonic Series (請預讀P259~P260)
However, nun=10.  divergent. (根據上一頁)
Prove:
Regrouping:
However,
Although when n, lnn, lnn14 when n=106. It diverges quite slowly.

12. Addition and multiplication of Series (請預讀P260~P261)
Exercises: (You need the concept of triangle inequality)
Hint: Use the definition of convergence to prove them.
If Snuns(u) and Snvns(v),
we will get Sn (un vn)s(u)  s(v).
If Snuns(u), we will get Sna*una*s(u).
Hint: Use Cauchy criterion to prove it.
If un0, Snuns(u) and {cn} are bounded, Snuncn is convergent.
課本漏了un0 的條件。
反例：un=(-1)n/n, cn=Q((-1)n), Snuncn is divergent.

13. Example: Oscillatory Series (請預讀P261)
You will find
Therefore, it is not convergent but oscillatory.
However,
It means that …(-1)n+…= ½.
Unfortunately, such correspondence between series and function is not unique and this approach must be refined.

14. Series discussed in this chapter
Riemann z-function: (P266)
Harmonic series: (P259)
Power series: (P291)
Geometric series: (P258)
Alternating series: (P270)

15. Homework
5.1.1
5.1.2
5.1.3

16. 5.1 nouns
Sequence (序列): A sequence of numbers (real or complex) is an ordered set of numbers in one-to-one correspondence with the positive integers; write {un}{u1, u2, … }.
Series (級數)： The sum of terms of a sequence:
Monotone:
Increasing (decreasing): un<un+1 (un > un+1)
Nondecreasing (nonincreasing): un  un+1 (un un+1)
Triangle inequality: |a|-|b| |a+b||a|+|b|
Mathematical induction (數學歸納法)