11.9 Representations offunctions as power series We already know that                =   + +        +   + ···        = con...
Ex: Find a power series representation for                                             +    and the radius of convergence.
Ex: Find a power series representation for                                             +    and the radius of convergence....
Ex: Find a power series representation for                                                   +    and the radius of conver...
Ex: Find a power series representation for                                                          +    and the radius of...
Ex: Find a power series representation for                                                             +    and the radius...
Ex: Find a power series representation for    and the radius of convergence.                                             +
Ex: Find a power series representation for    and the radius of convergence.                                             +...
Ex: Find a power series representation for    and the radius of convergence.                                              ...
Ex: Find a power series representation for    and the radius of convergence.                                              ...
Ex: Find a power series representation for    and the radius of convergence.                                              ...
Ex: Find a power series representation for    and the radius of convergence.                                             +
Ex: Find a power series representation for    and the radius of convergence.                                              ...
Ex: Find a power series representation for    and the radius of convergence.                                              ...
Ex: Find a power series representation for    and the radius of convergence.                                              ...
Ex: Find a power series representation for    and the radius of convergence.                                              ...
Ex: Find a power series representation for   (   )    and the radius of convergence.
Ex: Find a power series representation for   (   )    and the radius of convergence.   (     )=
Ex: Find a power series representation for   (   )    and the radius of convergence.   (     )=          =        + +     ...
Ex: Find a power series representation for   (   )    and the radius of convergence.   (     )=          =        + +     ...
Ex: Find a power series representation for   (   )    and the radius of convergence.   (     )=          =        + +     ...
Ex: Find a power series representation for   (   )    and the radius of convergence.   (     )=            =       + +    ...
Ex: Find a power series representation for     (   )    and the radius of convergence.   (     )=            =       + +  ...
Theorem (term-by-term diff. and int.):If the power series   ( − ) has a radiusof convergence , then                       ...
Ex: Find a power series representation for                                             (   )    and the radius of converge...
Ex: Find a power series representation for                                             (   )    and the radius of converge...
Ex: Find a power series representation for                                             (   )    and the radius of converge...
Ex: Find a power series representation for                                             (   )    and the radius of converge...
Ex: Find a power series representation for                                             (   )    and the radius of converge...
Ex: Find a power series representation for                                             (   )    and the radius of converge...
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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 .

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