Taylor_and_Maclaurin_Series_Calculus.ppt

Calculus

1. TAYLOR AND MACLAURIN
how to represent certain types of functions as sums of power series
You might wonder why we would ever want to express a known function
as a sum of infinitely many terms.
Integration. (Easy to integrate polynomials)
Finding limit
Finding a sum of a series (not only geometric, telescoping)
dxe
x
2 2
0
1
lim
x
xe
x
x



2. TAYLOR AND MACLAURIN
Example:x
exf)( 



0n
n
n
x
xce 
5
5
4
4
3
3
2
210 xcxcxcxcxcc
Maclaurin series ( center is 0 )
Example:xxf cos)( Find Maclaurin series

3. TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of Convergence
MEMORIZE: these Maclaurin Series

4. TAYLOR AND MACLAURIN
Maclaurin series ( center is 0 )
Example:xxf
1
tan)(


Find Maclaurin series

5. TAYLOR AND MACLAURIN
TERM-081

6. TAYLOR AND MACLAURIN
TERM-091

7. TAYLOR AND MACLAURIN
TERM-101

8. : TAYLOR AND MACLAURIN
TERM-082)2cos(cos
2
1
2
12
xx

9. Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
TERM-102

10. Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
TERM-091

11. TAYLOR AND MACLAURIN
Maclaurin series ( center is 0 )
Example:

0!
1
nn
Find the sum of the series

12. TAYLOR AND MACLAURIN
TERM-102

13. TAYLOR AND MACLAURIN
TERM-082

14. TAYLOR AND MACLAURIN
Leibniz’s formula:
Example: Find the sum 






0
12
1
12
)1()(tan
n
n
n
n
x
x 

753
)(tan
753
1 xxx
xx 



0 12
)1(
n
n
n 
7
1
5
1
3
1
1

15. TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of Convergence
MEMORIZE: these Maclaurin Series

16. The Binomial Series
Example:








3
3/1 !3
)2
3
1
)(1
3
1
(
3
1
 81
5

DEF:6
)
3
5
)(
3
2
(
3
1


Example:








5
2/1 !5
)4
2
1
)(3
2
1
)(2
2
1
)(1
2
1
(
2
1


17. The Binomial Series
binomial series.
NOTE:1
0








k k
kk









!11 !2
)1(
!22









 kkkk

18. The Binomial Series
TERM-101
binomial series.

19. The Binomial Series
TERM-092
binomial series.

20. TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of Convergence
Example:)1ln()( xxf  Find Maclaurin series

21. TAYLOR AND MACLAURIN
TERM-102

22. TAYLOR AND MACLAURIN
TERM-111

23. TAYLOR AND MACLAURIN
TERM-101

24. TAYLOR AND MACLAURIN
TERM-082

25. TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of Convergence
MEMORIZE: these Maclaurin Series

26. TAYLOR AND MACLAURIN
Maclaurin series ( center is 0)
Taylor series ( center is a)

27. TAYLOR AND MACLAURIN
TERM-091

28. TAYLOR AND MACLAURIN
TERM-092

29. TAYLOR AND MACLAURIN
TERM-082

30. TAYLOR AND MACLAURIN
Taylor series ( center is a)
Taylor polynomial of order n
DEF:

31. TAYLOR AND MACLAURIN
TERM-102
The Taylor polynomial of order 3 generated by the function f(x)=ln(3+x) at a=1 is:
Taylor polynomial of order n
DEF:

32. TAYLOR AND MACLAURIN
TERM-101

33. TAYLOR AND MACLAURIN
TERM-081

34. TAYLOR AND MACLAURIN
Taylor series ( center is a)



0
)(
)(
!
)(
)(
k
k
k
ax
k
af
xf
Taylor polynomial of order n


n
k
k
k
n ax
k
af
xP
0
)(
)(
!
)(
)(
Remainder 



1
)(
)(
!
)(
)(
nk
k
k
n ax
k
af
xR
Taylor Series )()()( xRxPxf
nn
Remainder consist of infinite terms k
n
n ax
n
cf
xR )(
)!1(
)(
)(
)1(




for some c between a and x.
Taylor’s Formula
REMARK:)(not )(
)1()1(
afcf
nn  Observe that :

35. TAYLOR AND MACLAURIN k
n
n ax
n
cf
xR )(
)!1(
)(
)(
)1(




for some c between a and x.
Taylor’s Formulak
n
n x
n
cf
xR
)!1(
)(
)(
)1(



for some c between 0 and x.
Taylor’s Formula

36. TAYLOR AND MACLAURIN
Taylor series ( center is a)
nth-degree Taylor polynomial of f at a.
DEF:
Remainder
DEF:)()()( xTxfxR
nn 
Example:





01
1
)(
n
n
x
x
xf 32
3
0
3
1)( xxxxxT
n
n

 


654
4
3
)( xxxxxR
n
n

37. TAYLOR AND MACLAURIN
TERM-092

38. TAYLOR AND MACLAURIN
TERM-081