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# Calculus II - 27

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Stewart Calculus 11.9

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• ### Calculus II - 27

1. 1. 11.9 Representations offunctions as power series We already know that = + + + + ··· = converges to when < < . On the contrary, can be expressed a power series = + + + + ··· = when < < .
2. 2. Ex: Find a power series representation for + and the radius of convergence.
3. 3. Ex: Find a power series representation for + and the radius of convergence.When | |< , = + + + ··· = =
4. 4. Ex: Find a power series representation for + and the radius of convergence.When | |< , = + + + ··· = =Therefore, = ( ) = ( ) + = =
5. 5. Ex: Find a power series representation for + and the radius of convergence.When | |< , = + + + ··· = =Therefore, = ( ) = ( ) + = =It converges when | |< , i.e. < < .
6. 6. Ex: Find a power series representation for + and the radius of convergence.When | |< , = + + + ··· = =Therefore, = ( ) = ( ) + = =It converges when | |< , i.e. < < .The radius of convergence is .
7. 7. Ex: Find a power series representation for and the radius of convergence. +
8. 8. Ex: Find a power series representation for and the radius of convergence. +When | |< , = + + + ··· = =
9. 9. Ex: Find a power series representation for and the radius of convergence. +When | |< , = + + + ··· = =Therefore, ( ) = = = + + + = =
10. 10. Ex: Find a power series representation for and the radius of convergence. +When | |< , = + + + ··· = =Therefore, ( ) = = = + + + = =It converges when | / |< , i.e. < < .
11. 11. Ex: Find a power series representation for and the radius of convergence. +When | |< , = + + + ··· = =Therefore, ( ) = = = + + + = =It converges when | / |< , i.e. < < .The radius of convergence is .
12. 12. Ex: Find a power series representation for and the radius of convergence. +
13. 13. Ex: Find a power series representation for and the radius of convergence. + ( ) = = = + + + = =
14. 14. Ex: Find a power series representation for and the radius of convergence. + ( ) = = = + + + = =It converges when | / |< , i.e. < < .
15. 15. Ex: Find a power series representation for and the radius of convergence. + ( ) = = = + + + = =It converges when | / |< , i.e. < < .The radius of convergence is .
16. 16. Ex: Find a power series representation for and the radius of convergence. + ( ) = = = + + + = =Therefore, ( ) ( ) + = + = + + = =It converges when | / |< , i.e. < < .The radius of convergence is .
17. 17. Ex: Find a power series representation for ( ) and the radius of convergence.
18. 18. Ex: Find a power series representation for ( ) and the radius of convergence. ( )=
19. 19. Ex: Find a power series representation for ( ) and the radius of convergence. ( )= = + + + + ···
20. 20. Ex: Find a power series representation for ( ) and the radius of convergence. ( )= = + + + + ··· = ··· +
21. 21. Ex: Find a power series representation for ( ) and the radius of convergence. ( )= = + + + + ··· = ··· + = + =
22. 22. Ex: Find a power series representation for ( ) and the radius of convergence. ( )= = + + + + ··· = ··· + = + = Take = , we get = .
23. 23. Ex: Find a power series representation for ( ) and the radius of convergence. ( )= = + + + + ··· = ··· + = + = Take = , we get = . The radius of convergence is .
24. 24. Theorem (term-by-term diff. and int.):If the power series ( − ) has a radiusof convergence , then ∞ ( )= = ( − )is differentiable on the interval ( − , + )and ∞ − ( )= = ( − ) + ∞ ( − ) ( ) = + = +The radii of both series are .
25. 25. Ex: Find a power series representation for ( ) and the radius of convergence.
26. 26. Ex: Find a power series representation for ( ) and the radius of convergence. = ( )
27. 27. Ex: Find a power series representation for ( ) and the radius of convergence. = ( ) =( + + + + ···)
28. 28. Ex: Find a power series representation for ( ) and the radius of convergence. = ( ) =( + + + + ···) = + + + ···
29. 29. Ex: Find a power series representation for ( ) and the radius of convergence. = ( ) =( + + + + ···) = + + + ··· = ( + ) =
30. 30. Ex: Find a power series representation for ( ) and the radius of convergence. = ( ) =( + + + + ···) = + + + ··· = ( + ) = The radius of convergence is .