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Calculus II - 28

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Stewart Calculus 11.10

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• Calculus II - 28

1. 1. 11.10 Taylor Series = = + + + ··· =Question: How (and whether) can we representa general function by power series? ( )= + ( )+ ( ) + ( ) + ···How to find , , , ··· ?
2. 2. ( )= + ( )+ ( ) + ( ) + ···How to determine ? = ( )How to determine ?( )= + ( )+ ( ) + ··· = ( )How to determine ?( )= + · ( )+ · ( ) + ··· = ( )/How to determine ? ( )= · + · · ( ) + ··· = ( )/ !
3. 3. Theorem: If ( ) has a power series representationat : ( )= ( ) , | |< =then its coefficients are given by the formula ( ) ( ) = !The series ( ) ( ) ( )( )= ( )+ ( )+ ( ) + ( ) + ··· ! ! !is called the Taylor series of the function at .
4. 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. 5. Ex: Find the Maclaurin series of ( )= andits radius of convergence.
6. 6. Ex: Find the Maclaurin series of ( )= andits radius of convergence. ( )= ( )= ( )= ···
7. 7. Ex: Find the Maclaurin series of ( )= andits radius of convergence. ( )= ( )= ( )= ( )= ( )= ( )= ··· ···
8. 8. Ex: Find the Maclaurin series of ( )= andits radius of convergence. ( )= ( )= ( )= ( )= ( )= ( )= ··· ··· ( ) ( ) = = + + + + ···= ! = ! ! ! !
9. 9. Ex: Find the Maclaurin series of ( )= andits radius of convergence. ( )= ( )= ( )= ( )= ( )= ( )= ··· ··· ( ) ( ) = = + + + + ···= ! = ! ! ! ! + | | = +
10. 10. Ex: Find the Maclaurin series of ( )= andits radius of convergence. ( )= ( )= ( )= ( )= ( )= ( )= ··· ··· ( ) ( ) = = + + + + ···= ! = ! ! ! ! + | | = + =
11. 11. Can we say that = ? Not yet. = !We need to prove that . = !
12. 12. Can we say that = ? Not yet. = !We need to prove that . = ! ( ) ( )Let = ( ) be the n-th degree = !Taylor polynomial of ( ) at . We need toprove ( )= ( ) ( ) .
13. 13. Taylor’s Inequality: ( + )If ( ) for | | , then theremainder of the Taylor series satisfies theinequality +| ( )| | | , | | ( + )!
14. 14. Why = ? = !
15. 15. Why = ? = ! ( + )Since ( ) for | |
16. 16. Why = ? = ! ( + )Since ( ) for | |by Taylor’s inequality, + | ( )| | | , | | ( + )!
17. 17. Why = ? = ! ( + )Since ( ) for | |by Taylor’s inequality, + | ( )| | | , | | ( + )! + | | = ( + )!
18. 18. Why = ? = ! ( + )Since ( ) for | |by Taylor’s inequality, + | ( )| | | , | | ( + )! + | | = ( + )!So ( )= , that is to say, = = !