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Understanding Limits and Convergence in Infinite Series and Sequences

This chapter provides a comprehensive overview of key concepts related to limits, convergence, and divergence within the context of sequences and series. It covers definitions such as the limit of a sequence, monotonic and bounded sequences, and properties of convergent and divergent series. The chapter also introduces essential convergence tests, including the Squeeze Theorem, Alternating Series Test, and Ratio Test. Additionally, it explores Taylor and Maclaurin series, their convergence, and operations with power series, equipping readers with the tools needed to analyze and understand infinite series.

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Understanding Limits and Convergence in Infinite Series and Sequences

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Presentation Transcript

  1. Chapter 1 Infinite Series

  2. Definition of the Limit of a Sequence

  3. Limit of a Sequence

  4. Properties of Limits of Sequences

  5. Squeeze Theorem for Sequences

  6. Absolute Value Theorem

  7. Definition of a Monotonic Sequence

  8. Definition of a Bounded Sequence

  9. Bounded Monotonic Sequences

  10. Definitions of Convergent and Divergent Series

  11. Convergence of a Geometric Series

  12. Properties of Infinite Series

  13. Limit of nth Term of a Convergent Series

  14. nth-Term Test for Divergence

  15. The Integral Test

  16. Convergence of p-Series

  17. Direct Comparison Test

  18. Limit Comparison Test

  19. Alternating Series Test

  20. Alternating Series Remainder

  21. Absolute Convergence

  22. Definitions of Absolute and Conditional Convergence

  23. Ratio Test

  24. Root Test

  25. Guidelines for Testing a Series for Convergence or Divergence

  26. Summary of Tests for Series

  27. Summary of Tests for Series (cont’d)

  28. Definitions of nth Taylor Polynomial and nth Maclaurin Polynomial

  29. Taylor's Theorem

  30. Definition of Power Series

  31. Convergence of a Power Series

  32. Properties of Functions Defined by Power Series

  33. Operations with Power Series

  34. The Form of a Convergent Power Series

  35. Definitions of Taylor and Maclaurin Series

  36. Convergence of Taylor Series

  37. Guidelines for Finding a Taylor Series

  38. Power Series for Elementary Functions

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