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Previous series have consisted of constants.

Section 10.7 – P ower Series. Previous series have consisted of constants. . Another type of series will include the variable x . . Section 10.7 – P ower Series. A Power Series is of the form. where x is a variable and represents the coefficients.

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Previous series have consisted of constants.

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  1. Section 10.7 – Power Series Previous series have consisted of constants.  Another type of series will include the variable x. 

  2. Section 10.7 – Power Series A Power Series is of the form where x is a variable and represents the coefficients. The sum of the series is the function where its domain is the set of all x for which the series converges.

  3. Section 10.7 – Power Series A more general form of the power series is of the form where x is a variable, represents the coefficients and a is a number. This form is referred as: a power series in (x – a) or a power series centered at a.

  4. Section 10.7 – Power Series Definitions The interval of x values where the series converges. Interval of Convergence: Half the length of the interval of convergence. Radius of Convergence (R): There are only three ways for a power series to converge. 1) The series only converges at . () () 2) The series converges for all x values. 3) The series converges for some interval of x. () The end values of the interval must be tested for convergence. The use of the Ratio test is recommended when finding the radius of convergence and the interval of convergence.

  5. Section 10.7 – Power Series Example: • Find the Radius of Convergence and the Interval of Convergence for the following power series  Ratio Test

  6. Section 10.7 – Power Series  Ratio Test: Convergence for Interval of Convergence: Radius of Convergence End points need to be tested.

  7. Section 10.7 – Power Series

  8. Section 10.7 – Power Series

  9. Section 10.7 – Power Series Example: • Find the Radius of Convergence and the Interval of Convergence for the following power series  Ratio Test

  10. Section 10.7 – Power Series  Ratio Test: Convergence for Interval of Convergence: Radius of Convergence End points need to be tested.

  11. Section 10.7 – Power Series

  12. Section 10.7 – Power Series

  13. Section 10.7 – Power Series Example: • Find the Radius of Convergence and the Interval of Convergence for the following power series  Ratio Test

  14. Section 10.7 – Power Series Example: • Find the Radius of Convergence and the Interval of Convergence for the following power series  Ratio Test

  15. Section 10.7 – Power Series Example: • Find the Radius of Convergence and the Interval of Convergence for the following power series Root Test 

  16. Section 10.7 – Power Series • Test the Endpoints 

  17. Section 10.7 – Power Series • Test the Endpoints 

  18. Section 10.7 – Power Series

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