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# Calculus II - 29

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Stewart Calculus 11.10

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• ### Calculus II - 29

1. 1. 11.10 Taylor SeriesTheorem: If ( ) has a power series representationat : ( )= ( ) , | |< =then its coefficients are given by the formula ( ) ( ) = !
2. 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. 3. Taylor’s Inequality: ( + )If ( ) for | | , then theremainder of the Taylor series satisfies theinequality +| ( )| | | , | | ( + )!
4. 4. Find the Maclaurin series for andprove that it represents for all .
5. 5. Find the Maclaurin series for andprove that it represents for all . ( )= ( )= ( )= ( )= ( )= ···
6. 6. Find the Maclaurin series for andprove that it represents for all . ( )= ( )= ( )= ( )= ( )= ( )= ( )= ( )= ( )= ( )= ··· ···
7. 7. Find the Maclaurin series for andprove that it represents for all .
8. 8. Find the Maclaurin series for andprove that it represents for all . ( ) ( ) ( ) ( )+ + + + ··· ! ! !
9. 9. Find the Maclaurin series for andprove that it represents for all . ( ) ( ) ( ) ( )+ + + + ··· ! ! ! = + + ··· ! ! !
10. 10. Find the Maclaurin series for andprove that it represents for all . ( ) ( ) ( ) ( )+ + + + ··· ! ! ! = + + ··· ! ! ! + = ( ) = ( + )!
11. 11. Find the Maclaurin series for andprove that it represents for all . ( ) ( ) ( ) ( )+ + + + ··· ! ! ! = + + ··· ! ! ! + = ( ) = ( + )!The radius of convergence is .
12. 12. Find the Maclaurin series for andprove that it represents for all .
13. 13. Find the Maclaurin series for andprove that it represents for all . ( + )Since ( )=± ,±
14. 14. Find the Maclaurin series for andprove that it represents for all . ( + )Since ( )=± ,± ( + )we know that | ( )| for all
15. 15. Find the Maclaurin series for andprove that it represents for all . ( + )Since ( )=± ,± ( + )we know that | ( )| for all +| ( )| | | for all ( + )!
16. 16. Find the Maclaurin series for andprove that it represents for all . ( + )Since ( )=± ,± ( + )we know that | ( )| for all +| ( )| | | for all ( + )!so + = ( ) = ( + )!
17. 17. Important Maclaurin Series: ∞ − = = = + + + ··· = ∞ = = ! = + ! + ! + ··· =∞ ∞ + = = (− ) ( + )! = ! − ! + ! − ! + ··· =∞ ∞ = = (− ) ( )! = − ! + ! − ! + ··· =∞ ∞ + − = = = (− ) + = − + − + ··· ∞( + ) = = ( − ) ( − )( − ) = + ! + ! + ! ··· =
18. 18. Derive more power series from the fundamental series.
19. 19. Derive more power series from the fundamental series. ∞ = = (− ) ( )! = − ! + ! − ! + ···
20. 20. Derive more power series from the fundamental series. ∞ = = (− ) ( )! = − ! + ! − ! + ··· ∞ ∞ + = = (− ) ( )! = = (− ) ( )!
21. 21. Derive more power series from the fundamental series. ∞ = = (− ) ( )! = − ! + ! − ! + ··· ∞ ∞ + = = (− ) ( )! = = (− ) ( )! ∞ = = ! = + ! + ! + ···
22. 22. Derive more power series from the fundamental series. ∞ = = (− ) ( )! = − ! + ! − ! + ··· ∞ ∞ + = = (− ) ( )! = = (− ) ( )! ∞ = = ! = + ! + ! + ··· − ∞ (− ) ∞ = = ! = = (− ) !
23. 23. Derive more power series from the fundamental series. ∞ = = (− ) ( )! = − ! + ! − ! + ··· ∞ ∞ + = = (− ) ( )! = = (− ) ( )! ∞ = = ! = + ! + ! + ··· − ∞ (− ) ∞ = = ! = = (− ) ! = + ! + ! + ··· · − ! + ! − ! + ···
24. 24. Derive more power series from the fundamental series. ∞ = = (− ) ( )! = − ! + ! − ! + ··· ∞ ∞ + = = (− ) ( )! = = (− ) ( )! ∞ = = ! = + ! + ! + ··· − ∞ (− ) ∞ = = ! = = (− ) ! = + ! + ! + ··· · − ! + ! − ! + ··· = + − − + ···
25. 25. Derive more power series from the fundamental series. ∞ = = (− ) ( )! = − ! + ! − ! + ··· ∞ ∞ + = = (− ) ( )! = = (− ) ( )! ∞ = = ! = + ! + ! + ··· − ∞ (− ) ∞ = = ! = = (− ) ! = + ! + ! + ··· · − ! + ! − ! + ··· = + − − + ··· = + ! + ! + ··· − ! + ! − ! + ···
26. 26. Derive more power series from the fundamental series. ∞ = = (− ) ( )! = − ! + ! − ! + ··· ∞ ∞ + = = (− ) ( )! = = (− ) ( )! ∞ = = ! = + ! + ! + ··· − ∞ (− ) ∞ = = ! = = (− ) ! = + ! + ! + ··· · − ! + ! − ! + ··· = + − − + ··· = + ! + ! + ··· − ! + ! − ! + ··· = + + + + ···