The document discusses Taylor series and Maclaurin series. It provides the formulas for determining the coefficients of a Taylor series centered at a point a. It also discusses Taylor's inequality which relates the remainder of a Taylor series to the derivative of the function. Several examples are worked out of finding the Maclaurin series for common functions and proving they represent the functions for all values. Additional power series are derived from fundamental series like e^x, sin(x), and cos(x).

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.
∞
= = (− ) ( )! = − ! + ! − ! + ···
∞ ∞ +
= = (− ) ( )! = = (− ) ( )!
∞
= = ! = + ! + ! + ···
− ∞ (− ) ∞
= = ! = = (− ) !
= + ! + ! + ··· · − ! + ! − ! + ···
= + − − + ···
= + ! + ! + ··· − ! + ! − ! + ···
= + + + + ···