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9.4

Radius of Convergence. 9.4. “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:. n th term test for divergence.

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9.4

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  1. Radius of Convergence 9.4

  2. “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:

  3. 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. Does the series converge or diverge? Note that this can prove that a series diverges, but can not prove that a series converges.

  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. NOTE: The nth term test does NOT prove convergence, it only proves divergence!

  5. 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.

  6. Direct Comparison Test This series converges. For non-negative series: So this series must also converge. If every term of a series is less than the corresponding term of a convergent series, then both series converge. So this series must also diverge. If every term of a series is greater than the corresponding term of a divergent series, then both series diverge. This series diverges.

  7. Does the series converge or diverge? If not for the + 2, this would be a geometric series with r = 4/3 which diverges. Since the + 2 is in the numerator, this series is larger than .

  8. 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.

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

  10. Does the series converge or diverge?

  11. What is the interval of convergence?

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

  13. The interval of convergence is (0,2). The radius of convergence is 1. Just a note: Watch the effect on the radius…

  14. The interval of convergence is (0,1). The radius of convergence is 1/2. Just a note: Watch the effect on the radius…

  15. The interval of convergence is (1/3,1). The radius of convergence is 1/3. Just a note: Watch the effect on the radius…

  16. Radius of convergence = 0. for all . At , the series is , which converges to zero. Ex: Note: If R is infinite, then the series converges for all values of x.

  17. Homework Page 495 #1-15, 17-33 odd On 17-33, also give interval of convergence.

  18. Another series for which it is easy to find the sum is the telescoping series. Using partial fractions: Ex. 6: p

  19. converges if , diverges if . p-series Test If this test seems backward after the ratio and nth root tests, remember that larger values of p would make the denominators increase faster and the terms decrease faster.

  20. Does the series converge or diverge? converge converge diverge diverge converge diverge

  21. It diverges very slowly, but it diverges. Because the p-series is so easy to evaluate, we use it to compare to other series. the harmonic series: diverges. (It is a p-series with p=1.)

  22. Homework Page 495 #2-12, 20-34 even, 35-39 (Geometric), 45,46,49,50

  23. If converges, then we say converges absolutely. If converges, then converges. Absolute Convergence The term “converges absolutely” means that the series formed by taking the absolute value of each term converges. Sometimes in the English language we use the word “absolutely” to mean “really” or “actually”. This is not the case here! If the series formed by taking the absolute value of each term converges, then the original series must also converge. “If a series converges absolutely, then it converges.”

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