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AP Calculus BC Monday, 14 April 2014

AP Calculus BC Monday, 14 April 2014. OBJECTIVE TSW (1) find a geometric power series that represents a function, and (2) Construct a power series using series operations. Calendars are not ready yet. ASSIGNMENTS DUE WEDNESDAY (write these down) Sec. 9.7 Sec. 9.8

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AP Calculus BC Monday, 14 April 2014

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  1. AP Calculus BCMonday, 14 April 2014 • OBJECTIVETSW (1) find a geometric power series that represents a function, and (2) Construct a power series using series operations. • Calendars are not ready yet. • ASSIGNMENTS DUE WEDNESDAY (write these down) • Sec. 9.7 • Sec. 9.8 • Sec. 9.9: p. 674 (1-25 odd) • Sec. 9.10: p. 685 (1-10 all, 21-26 all, 29)

  2. Sec. 9.9: Representation of Functions by Power Series

  3. Sec. 9.9: Representation of Functions by Power Series Consider This resembles the sum of a geometric series. Here, a = 1 and r = x. We could write the series as This series represents f only on the interval (−1, 1). To represent f on another interval, you must use a different series!

  4. This will converge when Sec. 9.9: Representation of Functions by Power Series Suppose we center at −1.

  5. Ex: Find a power series representation for centered at 0. Sec. 9.9: Representation of Functions by Power Series

  6. Ex: Find a power series representation for centered at 1. Sec. 9.9: Representation of Functions by Power Series

  7. Sec. 9.9: Representation of Functions by Power Series Using different operations on power series allows us to develop power series for a variety of elementary functions.

  8. Sec. 9.9: Representation of Functions by Power Series These operations may change the interval of convergence for the resulting series.

  9. Ex: Find a power series, centered at 0, for Then find the IOC. Sec. 9.9: Representation of Functions by Power Series Use partial fractions to rewrite: Write each fraction as a power series:

  10. Ex: Find a power series, centered at 0, for Then find the IOC. Sec. 9.9: Representation of Functions by Power Series Add these two series:

  11. Ex: Find a power series, centered at 0, for Then find the IOC. Now find the IOC for Sec. 9.9: Representation of Functions by Power Series

  12. We previously found Ex: Find a power series for centered at 1. Then find the IOC. If x = 1 (center is at 1), then Sec. 9.9: Representation of Functions by Power Series Let's integrate this:

  13. Ex: Find a power series for centered at 1. Then find the IOC. Sec. 9.9: Representation of Functions by Power Series Finally, find the IOC:

  14. Ex: Find a power series for centered at 1. Then find the IOC. Sec. 9.9: Representation of Functions by Power Series Diverges (IT) Converges (AST)

  15. Ex: Find a power series for centered at 0. Then find the IOC. Sec. 9.9: Representation of Functions by Power Series

  16. Ex: Find a power series for centered at 0. Then find the IOC. Sec. 9.9: Representation of Functions by Power Series

  17. Ex: Find a power series for centered at 0. Then find the IOC. Sec. 9.9: Representation of Functions by Power Series

  18. Ex: Find a power series for centered at 0. Then find the IOC. Sec. 9.9: Representation of Functions by Power Series Converges (AST) Converges (AST)

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