1. Series: Guide to Investigating Convergence

2. Understanding the Convergence of a Series

3. A series converges to λ if the limit of the sequence of the partial sums of the series is equal to λ

4. Example (1)

7. Warning

8. The Sum of the series

9. Questions Check, whether the given series is convergent, and if convergent find the its sum

10. Example (2) Telescoping Series

12. Warning

13. Examples of this type of telescoping series A Convergent Telescoping Series

14. Solutions

17. Examples of this type of telescoping series A Divergent Telescoping Series

18. Questions I Check, whether the given series is convergent, and if convergent find its sum

19. Questions II Show that the following series is a telescoping series, and then determine whether it is convergent

22. Example (3)

24. Warning

25. Questions Check, whether the given series is convergent.

26. Algebra of Series Convergence

28. Questions

29. Divergence Test

30. Questions Check, whether the given series is convergent.

31. Convergence Tests

32. Convergence Tests for Series of Positive Terms 1. Comparison Test 2. Limit Comparison Test 3. Ratio Test 4. Root Test

33. The Comparison test

34. Examples

35. Example (1)

36. Solution

38. Definition Order of Magnitude of a Series

39. Question

40. The Limit Comparison test

41. Examples

42. Solution

44. The Ratio test

45. Examples

46. The Root test

47. Examples

48. Definition Alternating Series

49. Alternating Series Convergence Test

50. Example

51. Definition Absolute and Conditional Convergence

52. Example (1)

53. Example (2)

54. The Ratio test for Absolute Convergence

55. Examples: Investigate the absolute convergence of the following series