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

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

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

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, … }. http://en.wikipedia.org/wiki/Sequence A series: The sum of terms of a sequence. Partial sums http://en.wikipedia.org/wiki/Series_(mathematics)

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？

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

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.

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

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=?

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

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.

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.

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

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

Homework 5.1.1 5.1.2 5.1.3

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)