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11.3

Ratio Test and Radius of Convergence. 11.3. Find the function represented by the series and the interval of convergence and the radius of convergence. “Does this series converge, and if so, for what values of x does it converge?”. Convergence.

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11.3

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  1. Ratio Test and Radius of Convergence 11.3

  2. Find the function represented by the series and the interval of convergence and the radius of convergence.

  3. “Does this series converge, and if so, for what values of x does it converge?” Convergence The series that are of the most interest to us are those that converge. Today we will consider the question:

  4. nth term test for divergence diverges if fails to exist or is not zero. The first requirement of convergence is that the terms must approach zero. Note that this can prove that a series diverges, but can not prove that a series converges.

  5. nth term test for divergence diverges if fails to exist or is not zero. The first requirement of convergence is that the terms must approach zero. Note that this can prove that a series diverges, but can not prove that a series converges. NOTE: The nth term test does NOT prove convergence, it only proves divergence!

  6. where r = common ratio between terms converges when We have learned that a geometric series given by: Geometric series have a constant ratio between terms. Other series have ratios that are not constant. If the absolute value of the limit of the ratio between consecutive terms is less than one, then the series will converge.

  7. For , if then: if the series converges. if the series diverges. if the series may or may not converge. The Ratio Test

  8. Does the series converge or diverge? How could we have come to the conclusion more easily?

  9. Ex:

  10. The interval of convergence is (2,8). The radius of convergence is . The series converges when

  11. for all . Radius of convergence = 0. At , the series is , which converges to zero. Ex: Series converges only at x= 3.

  12. Radius of convergence = . What would it mean if the ratio test simplified to this: for all Series converges for all real numbers.

  13. 1 There is a positive number R such that the series diverges for but converges for . The series converges for every x. ( ) 2 3 The series converges at and diverges everywhere else. ( ) There are three possibilities for power series convergence. The series converges over some finite interval: (the interval of convergence). The series may or may not converge at the endpoints of the interval. (As in the previous example.) The number R is the radius of convergence.

  14. Homework Page 493 #1-27 odd

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