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 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. The Integral Test

23. Example (3)

25. Warning

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

27. Algebra of Series Convergence

29. Questions

30. Divergence Test

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

32. Convergence Tests

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

34. The Comparison test

35. Examples

36. Example (1)

37. Solution

39. Definition Order of Magnitude of a Series

40. Question

41. The Limit Comparison test

42. Examples

43. Solution

45. The Ratio test

46. Examples

47. The Root test

48. Examples

49. Definition Alternating Series

50. Alternating Series Convergence Test

51. Example

52. Definition Absolute and Conditional Convergence

53. Example (1)

54. Example (2)

55. The Ratio test for Absolute Convergence

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

57. More Examples on the Integral Test

58. Example (1)

59. Example (2)

60. Example (3)