Transcript Maclaurin Series

Lesson 7 Maclaurin Series

Maclaurin Series
f ( x)  
n 0
f
(0)
(1)
(2)
f
( n)
(0) n
x
n!
(3)
(0) 0 f (0) 1 f (0) 2 f (0) 3
x 
x 
x 
x 
0!
1!
2!
3!
 (4)
f (0) 4 f (5) (0) 5
f ( n ) (0) n
x 
x  .... 
x
4!
5!
n!
Lesson 7 Maclaurin Series
Maclaurin Series

f ( x)  
n 0
f
( n)
(0) n
x
n!
Special Series To Remember Forever!
2
3
4
5
n
x
x
x
x
x
x
e  1  x      ...   ...
2! 3! 4! 5!
n!
x3 x5 x 7 x9
(1)n x 2n1
 ...
sin x  x      ... 
3! 5! 7! 9!
(2n  1)!
x 2 x 4 x 6 x8
(1)n x 2 n
 ...
cos x  1      ... 
2! 4! 6! 8!
(2n)!
1
 1  x  x2  x3  x 4  x5  ...  x n  ...
1 x
Lesson 7 Maclaurin Series
Use the MacLaurin Series for arctan x to
find the elusive number π

Maclaurin Series
f ( x)  
Step 1: Compute the derivatives
f ( x)  arctan x
f
f '( x) 
f ''( x) 
f '''( x) 
1
1 x 2
2 x
(1 x 2 ) 2
2(3 x 2 1)
(1 x 2 )3
(iv )
( x) 
f (v) ( x) 
f
( n)
n 0
(0) n
x
n!
24 x (1 x 2 )
(1 x 2 ) 4
24(110 x 2 5 x 4 )
(1 x2 )5
Lesson 7 Maclaurin Series
Step 2: Evaluate the derivatives at x=0
f (0)  arctan 0  0
f '(0) 
1
1 02
f ''(0) 
2 0
(1 02 ) 2
f '''(0) 
2(3 02 1)
(1 02 )3
f
f
(iv )
(v)
(0) 
(0) 
1
 0
 2
24 0(102 )
(1 0 )
2 4
 0
24(110 02  5 04 )
(1 0 )
2 5
 24
Lesson 7 Maclaurin Series
Step 3: Find the Maclaurin Series for arctan

f ( x)  
n 0

f
(2)
f
( n)
(0) n
x
n!
(3)
f (0) (0) 0
f (1) (0) 1

x 
x
0!
1!
f (0) 3
(0) 2
f (4) (0) 4
x 
x 
x
3!
2!
4!
f (5) (0) 5

x
5!
f ( n ) (0) n
f (6) (0) 6
f (7) (0) 7
x

x

x .... 
n!
6!
7!
24 5
0 4
1 1
0 0
0 2 2 3
 x
 x  x  x 
x  x
5!
4!
1!
0!
2!
3!
n
0 8
(6!) 7
0 6
(

1)
(
n

1
)
!
x  x .... 
 x 
xn
8!
7!
6!
n!
Lesson 7 Maclaurin Series
Step 3: Find the Maclaurin Series for arctan

f ( x)  
n 0
f
( n)
(0) n
x
n!
24 5
 x
5!
So...arctan x 
2 3

x
3!
1 1
 x
1!
(6!) 7

x
7!
3
(1) n (n  1) ! n
.... 
x
n!
5
7
x x x x
3
5
7
9
2 n 1
x
n x
 ....  (1)
2n  1
9
Lesson 7 Maclaurin Series
Let’s use a third-degree Maclaurin Polynomial
for arctan to estimate π
The third-degree Maclaurin Polynomial is:
x3
arctan x  x 
3
Since tan(π/4)=1, arctan(1)= π/4

2
13


Soooo.....arctan 1  1 
4
3
3
8
2
 2 23 2.667
Soooo.....  4 
3
3
Lesson 7 Maclaurin Series
Let’s use a fifth-degree Maclaurin Polynomial
for arctan to estimate π
The fifth-degree Maclaurin Polynomial is:
x3
x5

arctan x  x 
3
5
Since tan(π/4)=1, arctan(1)= π/4
5
13  
13
1

Soooo.....arctan 1  1 

4
15
3
5
52
13
 3 157  3.467
Soooo..... 
4 
15
15
Lesson 7 Maclaurin Series
Let’s use a seventh-degree Maclaurin
Polynomial for arctan to estimate π
The seventh-degree Maclaurin Polynomial is:
x3
x5

arctan x  x 
3
5
x7

7
Since tan(π/4)=1, arctan(1)= π/4
5
7
13
76  
1
1
Soooo.....arctan 1  1 

 
4
3
5
7 105
76
305
94
 2 105
Soooo..... 
4 
 2.895
105
105
Lesson 7 Maclaurin Series
Let’s use a ninth-degree Maclaurin Polynomial
for arctan to estimate π
The ninnth-degree Maclaurin Polynomial is:
x3
x5

arctan x  x 
3
5
x7

7
x9

9
Since tan(π/4)=1, arctan(1)= π/4
13 15 17 19 263  
Soooo.....arctan 1  1     
4
3
5 7
9 315
263
1052
 3 107
Soooo..... 
4
 3.340
315
315
315
Taylor Term
Reciprocals
Approximation for pi
1
1
4
-3
-0.333333333
2.666666667
5
0.2
3.466666667
-7
-0.142857143
2.895238095
9
0.111111111
3.33968254
-11
-0.090909091
2.976046176
13
0.076923077
3.283738484
-15
-0.066666667
3.017071817
17
0.058823529
3.252365935
-19
-0.052631579
3.041839619
21
0.047619048
3.232315809
-23
-0.043478261
3.058402766
3.058402766+3.232315809

2
3.14535