Download Ch 5.1 Fundamental Concepts of Infinite Series

講者： 許永昌 老師
1
Road-map of Chapter 5

Sequence (序列)



Bounded
Convergence and divergence
Series (級數)

Convergence and divergence

Convergent test



Cauchy criterion
lim un=0.
For positive un






Comparison Test
Cauchy root Test
Ratio Test
Integral Test
Absolute and Conditional Convergence
Series of

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
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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
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Definition of Sequence and Series
 A sequence of numbers: an
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.
N
 Partial sums s N   u k
k 1
 http://en.wikipedia.org/wiki/Series_(mathematics)
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An example of sequence: damped
oscillation
 t=1,2,3,… (s)
 It is
Q: 怎麼文字表達它們？
 It is
sin(0.4*t)*exp(-0.01*t)+1
2.5
2
1.5
1
0.5
0
0
50
100
150
200
250
300
350
400
450
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Bounded and convergent
 Reference: Introduction to Mathematical Physics, M.T.
Vaughn
 The sequence {un} is bounded
.
 In this example, M=2.
 The sequence {un} is convergent to the
u
.
 In this case, u=1.
 N=250,e=0.2 for dotted lines.
un+1
un
u
e
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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.
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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.
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)+…
 P0 2,1  P0 2, A
1
P0  A,1
P -1  A - P0  A, A
2
 Reference: A Guide to Feynman
2
Diagrams in the Many-Body Problem.
2
2
=
1
A
+
A +
A
+…
1
1
1
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Sequence and Series

 P2,1  P 2,1  P 2, AP A P  A, AP A P  A,1
 If we define P0(A,A)P(A)=r, we get
0
0
k
0
k 0

0

k -1






P
A
,
A
P
A

r
 0

k
k 0
k 1
Which is an infinite series.

Sequence: {0.5n}
0.6
{S 0.5k}
1.2
0.5
0.4
Partial sums
0.3
1
0.8
0.6
0.2
S0.5^k
0.4
0.1
0.2
0
1
2
3
4
5
6
7
8
9
10
11
12
Converged? S=?
0
0
5
10
15
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Necessary condition for the
convergence of a series (請預讀P259)
 The necessary condition is lim u k  0.
k 
: Prove it.
 However, it is
to guarantee convergence.
 In Ch5.2 it will tell us the general 4 kinds of test.
 If the (i)
are (ii) monotonic decreasing to zero, that
is,
for
, then Snun is converging to S if, and
only if, sn-nun converges to S.
 Prove: Hint:

sn
tnsn-nun<sn.
sn

tn - sv  uv 1 
n
 u
m v  2
m
tn
- un  -  v  1 un  0
If
uv1
 un
v 1
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Example of divergent series:
Harmonic Series (請預讀P259~P260)

1
k 1 k
 Harmonic series : 
1
 0.
k  k
 lim
 However, nun=1 0.  divergent. (根據上一頁)
 Prove:
 Regrouping:

1
1 1  1 1 1 1

1

     2      ....

 2 3  2 5 6 7 
k 1 k
1 1
 1    ....  
2 -1
1
12 2
1
m1


k 2m
k

2
m 1


* 2 m 1 - 2 m 
2
 However, 1 1 dx  ln n.
x
 Although when n, lnn, lnn14 when n=106. It diverges
quite slowly.
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n
Addition and multiplication of
Series (請預讀P260~P261)
: (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.
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Example: Oscillatory Series (請預讀
P261)


k -1


1

k 1
 You will find
2n
 s2 n   - 1k -1  0
k 1
s2 n 1 
2 n 1
 - 1
k -1
1
k 1
 Therefore, it is not convergent but oscillatory.

1
k -1
 However,
  - x   1 - x  x 2 - ...
1  x k 1
 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.
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Series discussed in this chapter

 z s   
1
s
k 1 k
z-function: (P266)


series: (P259)
s  z 1  
1
k 1 k
series:
(P291)

k
 S  x    ak x
k 0


S x    x k
k 0

series: (P258)
series: (P270)
k
 S   - 1 uk , where uk  0.
k 0
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Homework
 5.1.1
 5.1.2
 5.1.3
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5.1 nouns
(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, … }.
(級數)： The sum of terms of a sequence: s   u .
:
N
N
(
(
k 1
k
): un<un+1 (un > un+1)
): un  un+1 (un  un+1)
: |a|-|b| |a+b||a|+|b|
(數學歸納法)
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