Stewart Calculus 11.10

1. 11.10 Taylor Series
Theorem: If ( ) has a power series representation
at :
( )= ( ) , | |<
=
then its coefficients are given by the formula
( )
( )
=
!

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

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

4. Find the Maclaurin series for and
prove that it represents for all .

5. Find the Maclaurin series for and
prove that it represents for all .
( )=
( )=
( )=
( )=
( )=
···

6. Find the Maclaurin series for and
prove that it represents for all .
( )= ( )=
( )= ( )=
( )= ( )=
( )= ( )=
( )= ( )=
··· ···

7. Find the Maclaurin series for and
prove that it represents for all .

8. Find the Maclaurin series for and
prove that it represents for all .
( ) ( ) ( )
( )+ + + + ···
! ! !

9. Find the Maclaurin series for and
prove that it represents for all .
( ) ( ) ( )
( )+ + + + ···
! ! !
= + + ···
! ! !

10. Find the Maclaurin series for and
prove that it represents for all .
( ) ( ) ( )
( )+ + + + ···
! ! !
= + + ···
! ! !
+
= ( )
=
( + )!

11. Find the Maclaurin series for and
prove that it represents for all .
( ) ( ) ( )
( )+ + + + ···
! ! !
= + + ···
! ! !
+
= ( )
=
( + )!
The radius of convergence is .

12. Find the Maclaurin series for and
prove that it represents for all .

13. Find the Maclaurin series for and
prove that it represents for all .
( + )
Since ( )=± ,±

14. Find the Maclaurin series for and
prove that it represents for all .
( + )
Since ( )=± ,±
( + )
we know that | ( )| for all

15. Find the Maclaurin series for and
prove that it represents for all .
( + )
Since ( )=± ,±
( + )
we know that | ( )| for all
+
| ( )| | | for all
( + )!

16. Find the Maclaurin series for and
prove that it represents for all .
( + )
Since ( )=± ,±
( + )
we know that | ( )| for all
+
| ( )| | | for all
( + )!
so
+
= ( )
=
( + )!

17. Important Maclaurin Series:
∞
− = = = + + + ··· =
∞
= = ! = + ! + ! + ··· =∞
∞ +
= = (− ) ( + )! = ! − ! + ! − ! + ··· =∞
∞
= = (− ) ( )! = − ! + ! − ! + ··· =∞
∞ +
− =
= = (− ) + = − + − + ···
∞
( + ) = =
( − ) ( − )( − )
= + ! + ! + ! ··· =

18. Derive more power series from the fundamental series.

19. Derive more power series from the fundamental series.
∞
= = (− ) ( )! = − ! + ! − ! + ···

20. Derive more power series from the fundamental series.
∞
= = (− ) ( )! = − ! + ! − ! + ···
∞ ∞ +
= = (− ) ( )! = = (− ) ( )!

21. Derive more power series from the fundamental series.
∞
= = (− ) ( )! = − ! + ! − ! + ···
∞ ∞ +
= = (− ) ( )! = = (− ) ( )!
∞
= = ! = + ! + ! + ···

22. Derive more power series from the fundamental series.
∞
= = (− ) ( )! = − ! + ! − ! + ···
∞ ∞ +
= = (− ) ( )! = = (− ) ( )!
∞
= = ! = + ! + ! + ···
− ∞ (− ) ∞
= = ! = = (− ) !

23. Derive more power series from the fundamental series.
∞
= = (− ) ( )! = − ! + ! − ! + ···
∞ ∞ +
= = (− ) ( )! = = (− ) ( )!
∞
= = ! = + ! + ! + ···
− ∞ (− ) ∞
= = ! = = (− ) !
= + ! + ! + ··· · − ! + ! − ! + ···

24. Derive more power series from the fundamental series.
∞
= = (− ) ( )! = − ! + ! − ! + ···
∞ ∞ +
= = (− ) ( )! = = (− ) ( )!
∞
= = ! = + ! + ! + ···
− ∞ (− ) ∞
= = ! = = (− ) !
= + ! + ! + ··· · − ! + ! − ! + ···
= + − − + ···

25. Derive more power series from the fundamental series.
∞
= = (− ) ( )! = − ! + ! − ! + ···
∞ ∞ +
= = (− ) ( )! = = (− ) ( )!
∞
= = ! = + ! + ! + ···
− ∞ (− ) ∞
= = ! = = (− ) !
= + ! + ! + ··· · − ! + ! − ! + ···
= + − − + ···
= + ! + ! + ··· − ! + ! − ! + ···

26. Derive more power series from the fundamental series.
∞
= = (− ) ( )! = − ! + ! − ! + ···
∞ ∞ +
= = (− ) ( )! = = (− ) ( )!
∞
= = ! = + ! + ! + ···
− ∞ (− ) ∞
= = ! = = (− ) !
= + ! + ! + ··· · − ! + ! − ! + ···
= + − − + ···
= + ! + ! + ··· − ! + ! − ! + ···
= + + + + ···