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Lecture 24 – Divergence Test

Lecture 24 – Divergence Test. Goal: Does a series converge or diverge?. Divergence Test (If a series converges, then sequence converges to 0.). Example 1 – Converge/Diverge?. Example 2 – Converge/Diverge?. Example 3 – Converge/Diverge?. However,. Example 4 – Converge/Diverge?.

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Lecture 24 – Divergence Test

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  1. Lecture 24 – Divergence Test Goal: Does a series converge or diverge? Divergence Test(If a series converges, then sequence converges to 0.)

  2. Example 1 – Converge/Diverge? Example 2 – Converge/Diverge?

  3. Example 3 – Converge/Diverge? However,

  4. Example 4 – Converge/Diverge? However,

  5. Lecture 25 – Integral Test Goal: Does a series converge or diverge? Integral Test(The area under a function and infinite sum of the terms in a sequence defined by that function are related.)

  6. If area under curve is bounded, then so is But then is a bounded, monotonic sequence. So it converges and thus converges. If area under curve is unbounded, then is also unbounded. And thus, diverges.

  7. Example 1 – Converge/Diverge? sequenceconvergesto zero. No info from Divergence Test.

  8. Example 2 – Converge/Diverge? sequenceconvergesto zero. No info from Divergence Test.

  9. Example 3 – Converge/Diverge? sequenceconvergesto zero. No info from Divergence Test.

  10. Example 4 – Converge/Diverge? sequenceconvergesto zero. No info from Divergence Test.

  11. Lecture 26 – Ratio and Root Tests Goal: Does a series of positive termsconverge or diverge? Ratio Test(Does ratio of successive terms approach some limit L? Then series is close to being geometric.)

  12. Root Test(Does nth root of terms approach some limit L? Then series is close to being geometric.)

  13. Example 1 – Converge/Diverge? sequenceconvergesto zero. No info from Divergence Test.

  14. Example 2 For what values k does the series converge?

  15. Example 3 – Converge/Diverge? sequenceconvergesto zero. No info from Divergence Test.

  16. Example 4 – Converge/Diverge?

  17. Lecture 27 – Comparison Tests 1. 2. Direct Comparison:

  18. Limit Comparison:

  19. Example 1 – Converge/Diverge? sequenceconvergesto zero. No info from Divergence Test.

  20. Example 2 – Converge/Diverge? sequenceconvergesto zero. No info from Divergence Test.

  21. Example 3 – Converge/Diverge? sequenceconvergesto zero. No info from Divergence Test.

  22. Example 4 – Converge/Diverge? sequenceconvergesto zero. No info from Divergence Test.

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