Stewart Calculus 11.10

1. 11.10 Taylor Series
= = + + + ···
=
Question: How (and whether) can we represent
a general function by power series?
( )= + ( )+ ( ) + ( ) + ···
How to find , , , ··· ?

2. ( )= + ( )+ ( ) + ( ) + ···
How to determine ?
= ( )
How to determine ?
( )= + ( )+ ( ) + ···
= ( )
How to determine ?
( )= + · ( )+ · ( ) + ···
= ( )/
How to determine ?
( )= · + · · ( ) + ···
= ( )/ !

3. Theorem: If ( ) has a power series representation
at :
( )= ( ) , | |<
=
then its coefficients are given by the formula
( )
( )
=
!
The series
( ) ( ) ( )
( )= ( )+ ( )+ ( ) + ( ) + ···
! ! !
is called the Taylor series of the function at .

4. The series
( ) ( ) ( )
( )= ( )+ ( )+ ( ) + ( ) + ···
! ! !
is called the Taylor series of the function at .
The special case when = :
( ) ( ) ( )
( )= ( )+ + + + ···
! ! !
is called the Maclaurin series of the function.

5. Ex: Find the Maclaurin series of ( )= and
its radius of convergence.

6. Ex: Find the Maclaurin series of ( )= and
its radius of convergence.
( )=
( )=
( )=
···

7. Ex: Find the Maclaurin series of ( )= and
its radius of convergence.
( )= ( )=
( )= ( )=
( )= ( )=
··· ···

8. Ex: Find the Maclaurin series of ( )= and
its radius of convergence.
( )= ( )=
( )= ( )=
( )= ( )=
··· ···
( )
( )
= = + + + + ···
=
! =
! ! ! !

9. Ex: Find the Maclaurin series of ( )= and
its radius of convergence.
( )= ( )=
( )= ( )=
( )= ( )=
··· ···
( )
( )
= = + + + + ···
=
! =
! ! ! !
+ | |
=
+

10. Ex: Find the Maclaurin series of ( )= and
its radius of convergence.
( )= ( )=
( )= ( )=
( )= ( )=
··· ···
( )
( )
= = + + + + ···
=
! =
! ! ! !
+ | |
=
+
=

11. Can we say that = ? Not yet.
=
!
We need to prove that .
=
!

12. Can we say that = ? Not yet.
=
!
We need to prove that .
=
!
( )
( )
Let = ( ) be the n-th degree
=
!
Taylor polynomial of ( ) at . We need to
prove ( )= ( ) ( ) .

13. Taylor’s Inequality:
( + )
If ( ) for | | , then the
remainder of the Taylor series satisfies the
inequality
+
| ( )| | | , | |
( + )!

14. Why = ?
=
!

15. Why = ?
=
!
( + )
Since ( ) for | |

16. Why = ?
=
!
( + )
Since ( ) for | |
by Taylor’s inequality,
+
| ( )| | | , | |
( + )!

17. Why = ?
=
!
( + )
Since ( ) for | |
by Taylor’s inequality,
+
| ( )| | | , | |
( + )!
+
| | =
( + )!

18. Why = ?
=
!
( + )
Since ( ) for | |
by Taylor’s inequality,
+
| ( )| | | , | |
( + )!
+
| | =
( + )!
So ( )= , that is to say, =
=
!