講者： 許永昌 老師
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Contents
 Bernoulli function
 Euler-Maclaurin Integration Formula
 Improvement of Convergence
 Asymptotic Series
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Bernoulli Functions (請預讀P305~P306)
 We use its properties to derive the Euler-Maclaurin
Integration Formula.
 Bernoulli Functions: Bn(s)

xe xs
xn
  Bn  s 
x
e  1 n 0
n!

 Properties:


Bernoulli numbers: Bn=Bn(0).
d
Bn  s   nBn 1  s 
ds
:
Prove them.
Bn 1   1 Bn  0 
n



B0(s)=1.
B2n+1=0, n  1.
3
Bernoulli functions (continue)
 From the properties of Bernoulli functions, we get
B0  x   1,
1
B1  x   x  .
2
 However, you will find that
B2  x   x 2  x  B2
3x 2
B3  x   x 
 3B2 x
2
 We can get B2
3
by the relation B3(1)=(-1)3B3=0, i.e.
1-3/2+3B2=0.

B2=1/6.
 當然，您也可以用課本 Eq. (5.124)來回答。
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Euler-Maclaurin Integration
Formula (請預讀P306~P307)
 Euler-Maclaurin Integration Formula

1
0
q
1
1
1 1 2q
f  x  dx   f 1  f  0    
B2 p  f  2 p 1 1  f  2 p 1  0   
f
 x  B2q  x  dx,
2
 2q ! 0
p 1  2 p  !
or

n
0
f  x  dx 
1
1
f  0   f 1  ...  f  n  1  f  n 
2
2
Trapezoidal integration (梯形法 )
1 n 1
1
1
2 p 1
2 p 1


2q

B2 p  f
f
n  f
 0  
 x  m B2 q  x  dx,


0
2
p
!
2
q
!

  m 0
p 1 
q
 Hint:

1
0
f  p   x  B p  x  dx 
dBn  x 
dx

1
dB p 1  x 
1
1 1  p
1
 p
 p 1
f
x
dx

f
x
B
x

f






 x  B p 1  x  dx
p

1
0
0
p  1 0
dx
P 1
 nBn 1  x  ,
B2 n 1  0, B2 n 1  B2 n  0  .
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
Example
 Sm2=?
 Hint:
Based on the Euler-Maclaurin Formula


n
0
f  x  dx 
1
1
f  0   f 1  ...  f  n  1  f  n 
2
2
Trapezoidal integration (梯形法 )
1 n 1
1
1
2 p 1
2 p 1


2q

B2 p  f
f
n  f
 0  
 x  m B2q  x  dx,


0
 2q  ! m  0
p 1  2 p  !
q


f(p)(x)=0 when p>2.
ln  x ! 
1
ln  2 
2
1

  x   ln x  x  O x 1 .
2

 
It will be discussed in Ch 7.
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Improvement of convergence (請
預讀P307~P311)
 If a series is

ak
m
k 1 k
s
, its ratio and root tests cannot
tell us whether it is converged and we need to find
another way to determine it.
: If ak=1, what is the condition for m so
that sn is a convergent series?
 How can we improve its rate of convergence?
 If the original series converges as km, we try to find a
way to make this series converges as km+p.
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Improvement of convergence
(continue)
 Method I:

 We know that a m1  
.
1
1

m  m!
k 1 k  k  1 ...  k  m 
 Find a constant c so that s= cam + D and D converges as
k(m+1).
 Method II:
s by (1+a1x)
 Find a constant a1 so that s= S/(1+a1x) and S converges as
k(m+1).
 Based on the same concepts, you can make the
convergence better than k(m+1).
 Method III? :
x  1/x when x is larger enough.
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Homework
 5.9.5
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An Example of Asymptotic Series (請
預讀P314~P316)
2
e dt ,
 Example: error function erf  x  

 0
 Power series:
n
n
n 2 n 1
2




t
x
1 x 2 n
1 x





erf  x    
dt  
t dt  
,

0
2
n!
n! 0
n 0
n 0
n  0 n ! 2n  1



x
t 2
Radius of convergence: R=
However, it
.
Therefore, people tried to find another kind of expansion to
solve this problem.
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An example of Asymptotic series
(continue)
 Another form (An asymptotic series)
2
erf  x  


x
 
0
x
e dt  1 

x

x

t 2
e dt   
t
2


x
t 2
 
 
x
t 2
e
e
d t 2  
2t
2t

t 2

t 2
e dt



x
x
 
t
e
et
e x
dt 

dt
2
2
x
2t
2x
2t
2
 x2
2

e
e
e
2
dt


d

t


x 2t 2n1
t 2n
2 x 2 n 1 x
e x
t
e dt 
2x
2
2
t 2
2
 2n  1 et
2t 2 n  2
2
2
dt

n 1
t 2
n

 e
 1  
 1 
1    2  1!! 2     2n  1!!  x 2 n  2 dt
t
 2 x  
 2 
  1
 Its infinite series diverges everywhere.
 Please check it by
.
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An example of Asymptotic series
(continue)
 Another way to get this result:


x
e t dt 
2
t  x


0 e
v m
v dv  m!
v
2x

 x2
e
2x

0
e


n 0
 v 
 x 2  v  
 2x 
2
 x2
d
v e

2x 2x
 1  2n !  e x
n
n !22 n x 2 n


0

ev 
n 0
 1
n
n ! 2 x 
2n
v 2 n dv
n



 1 
1    2   2n  1!!
2 x  n 1  2 x 

2
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An example of Asymptotic series
(continue)
 Although its
,
.

1
Rn 1  
 2n  1!! 

 2 
2


x
et
e x
dt  2 n  2
t 2n2
v x
2
2
t  x
 Rn 1

2x
2n  1!! e x




x
et
dt
t 2n2
2
 v 

 v  2 x 

0
n 1

e e
1 v
2 x2

2
2n2
v
e x
d

2 x 2 x 2 n 3
2

ev
 1
0
2n2
e x
dv  2 n 3
2x
2
2
2 n 3
2n 1  x
 Therefore, if we take x large enough, our partial sum is
an arbitrarily good approximation to the error function.
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Asymptotic Series
 Asymptotic Series:
 They are sometimes called a


.
以上例來看，x，
，fn(x)0，
但是，若是 x 固定， n時， fn(x) 。
 They are
and never infinite series.
 They are useful, often more useful than convergent
series.
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Homework
 5.10.6
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Nouns
Bn(x)
BnBn(0)
:
P305
P307
Ch5.10
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