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
x
e
 dx
2
ex 1  x
x 0
x2
lim
 Finding a sum of a series (not only geometric, telescoping)
TAYLOR AND MACLAURIN
Example:
f ( x)  e x

e   cn x n  c0  c1 x  c2 x 2  c3 x 3  c4 x 4  c5 x 5  
x
n 0
Maclaurin series ( center is 0 )
Example:
Find Maclaurin series
f ( x )  cos x
TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of Convergence
MEMORIZE: these Maclaurin Series
TAYLOR AND MACLAURIN
Maclaurin series ( center is 0 )
Example:
Find Maclaurin series
f ( x)  tan 1 x
TAYLOR AND MACLAURIN
TERM-081
TAYLOR AND MACLAURIN
TERM-091
TAYLOR AND MACLAURIN
TERM-101
: TAYLOR AND MACLAURIN
TERM-082
cos 2 x  12  12 cos(2 x)
Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
TERM-102
Sec 11.9 & 11.10: TAYLOR AND MACLAURIN
TERM-091
TAYLOR AND MACLAURIN
Maclaurin series ( center is 0 )
Example:
Find the sum of the series

1

n 0 n!
TAYLOR AND MACLAURIN
TERM-102
TAYLOR AND MACLAURIN
TERM-082
TAYLOR AND MACLAURIN
Example:
(1) n

n  0 2n  1

Find the sum
Leibniz’s formula:
 1
1 1 1
  
3 5 7
x 2 n1
tan ( x)   (1)
2n  1
n 0
1

n
x3 x5 x7
tan ( x)  x 



3
5
7
1
TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of Convergence
MEMORIZE: these Maclaurin Series
The Binomial Series
DEF:
Example:
 1/ 3 
  
3
1 1
1
( 1)(  2 )
3 3
3
3!

1 2
5
(  )(  )
3 3
3
6
Example:
 1/ 2 
  
5
1 1
1
1
1
( 1)(  2 )( 3)(  4 )
2 2
2
2
2
5!

5
81
The Binomial Series
binomial series.
NOTE:
k
   1
 0
k k
    k
1 1!
 k  k k (k  1)
   
2!
 2  2!
The Binomial Series
TERM-101
binomial series.
The Binomial Series
TERM-092
binomial series.
TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of Convergence
Example:
Find Maclaurin series
f ( x)  ln( 1  x)
TAYLOR AND MACLAURIN
TERM-102
TAYLOR AND MACLAURIN
TERM-111
TAYLOR AND MACLAURIN
TERM-101
TAYLOR AND MACLAURIN
TERM-082
TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of Convergence
MEMORIZE: these Maclaurin Series
TAYLOR AND MACLAURIN
Maclaurin series ( center is 0 )
Taylor series ( center is a )
TAYLOR AND MACLAURIN
TERM-091
TAYLOR AND MACLAURIN
TERM-092
TAYLOR AND MACLAURIN
TERM-082
TAYLOR AND MACLAURIN
Taylor series ( center is a )
DEF:
Taylor polynomial of order n
TAYLOR AND MACLAURIN
The Taylor polynomial of order 3 generated by the function f(x)=ln(3+x) at a=1 is:
TERM-102
DEF:
Taylor polynomial of order n
TAYLOR AND MACLAURIN
TERM-101
TAYLOR AND MACLAURIN
TERM-081
TAYLOR AND MACLAURIN

f ( x)  
k 0
n
Pn ( x)  
k 0
Rn ( x) 
f ( k ) (a)
( x  a) k
k!
Taylor series ( center is a )
f ( k ) (a)
( x  a) k
k!
Taylor polynomial of order n


k  n 1
f ( k ) (a)
( x  a) k
k!
f ( x)  Pn ( x)  Rn ( x)
Taylor’s Formula
Remainder
Taylor Series
f ( n1) (c)
Rn ( x) 
( x  a) k
(n  1)!
for some c between a and x.
REMARK:
Observe that :
f ( n1) (c) not f ( n1) (a)
Remainder consist of infinite terms
TAYLOR AND MACLAURIN
Taylor’s Formula
f ( n1) (c)
Rn ( x) 
( x  a) k
(n  1)!
for some c between a and x.
Taylor’s Formula
f ( n1) (c) k
Rn ( x) 
x
(n  1)!
for some c between 0 and x.
TAYLOR AND MACLAURIN
Taylor series ( center is a )
DEF:
nth-degree Taylor polynomial of f at a.
DEF:
Rn ( x)  f ( x)  Tn ( x)
Example:

1
f ( x) 
  xn
1  x n 0
Remainder
3
T3 ( x)   x n  1  x  x 2  x 3
n 0

R3 ( x)   x n  x 4  x 5  x 6  
n4
TAYLOR AND MACLAURIN
TERM-092
TAYLOR AND MACLAURIN
TERM-081