1. SERIES
DEF:
A sequence is a list of numbers written in a definite order:
DEF:
Is called a series
Example:

2. What is the difference:
SERIES
What is the difference:

3. SERIES DEF: DEF: Is called a series its sum n-th term convergent
Example:
its sum
n-th term
DEF:
If the sum of the series
convergent
is finite number not infinity

4. SERIES DEF: DEF: Given a seris nth-partial sums : Given a seris
Example:
DEF:
Given a seris
nth-partial sums :
DEF:
Given a seris
the sequence of partial sums. :

5. SERIES We define Given a series Given a series
Sequence of partial sums
Given a series
Sequence of partial sums

6. Sequence of partial sums
SERIES
We define
Given a series
Sequence of partial sums
DEF: If
convergent
convergent If
divergent
divergent

7. SERIES
Final-121

8. SERIES
Example:

9. SERIES Special Series: Geometric Series: Example Example
Harmonic Series
Telescoping Series
p-series
Alternating p-series
first term
Common ratio
Example
Example:
Example
Is it geometric?
Is it geometric?

10. SERIES Geometric Series: Geometric Series: Example Example Example:
Is it geometric?

11. SERIES
Final-111

12. SERIES
Final-102

13. SERIES
Final-121

14. SERIES
Geometric Series:
Geometric Series:
prove:

15. SERIES
Geometric Series:
Geometric Series:

16. SERIES
Final-102

17. SERIES
Geometric Series:
Geometric Series:

18. SERIES Special Series: Telescoping Series: Telescoping Series:
Geometric Series
Harmonic Series
Telescoping Series
p-series
Alternating p-series
Telescoping Series:
Convergent
Convergent
Example:
Remark:
b1 means the first term
( n starts from what integer)

19. SERIES
Telescoping Series:
Convergent
Convergent
Final-111

20. SERIES Telescoping Series: Telescoping Series: Convergent Convergent
Notice that the terms cancel in pairs. This is an example of a telescoping sum: Because of all the cancellations, the sum collapses (like a pirate’s collapsing telescope) into just two terms.

21. SERIES
Final-132

22. SERIES
Final-141

23. THEOREM: Convergent SERIES Example: Example: Example: divergent
In general, the converse is not true

24. SERIES
THEOREM:
Convergent
THEOREM:THE TEST FOR DIVERGENCE
Divergent

25. THEOREM:THE TEST FOR DIVERGENCE
SERIES
THEOREM:THE TEST FOR DIVERGENCE
Divergent
Example:
Is the series convergent or divergent?

26. SERIES THEOREM:THE TEST FOR DIVERGENCE Divergent THEOREM: Convergent
REMARK(1):
The converse of Theorem is not true in general. If we cannot conclude that
is convergent.
Convergent
REMARK(2):
the set of all series
If we find that
we know nothing about the convergence or divergence

27. SERIES THEOREM: Convergent Seq. series convg div REMARK(2): REMARK(3):
Sequence
REMARK(2):
REMARK(3):
convg
convg
REMARK(4):
div
div

28. SERIES
REMARK
Example
All these items are true if these two series are convergent

29. SERIES
Final-081

30. SERIES
Final-082

31. SERIES
Final-101
Final-112

32. Adding or Deleting Terms
SERIES
Adding or Deleting Terms
REMARK(4):
Example
A finite number of terms doesn’t affect the divergence of a series.
REMARK(5):
Example
A finite number of terms doesn’t affect the convergence of a series.
REMARK(6):
A finite number of terms doesn’t affect the convergence of a series but it affect the sum.

33. SERIES
Reindexing
Example
We can write this geometric series

34. SERIES Special Series: Geometric Series Harmonic Series
Telescoping Series
p-series
Alternating p-series

35. summary General Telescoping Geometric SERIES Divergent convg
When convg
sum
nth partial sum
THEOREM:THE TEST FOR DIVERGENCE
Divergent
convg
convg

38. Extra Problems

39. SERIES
Final-101

40. SERIES
Final-112

41. SERIES
Final-101

42. SERIES
Final-092

43. SERIES
Final-121

44. SERIES
Final-103

45. SERIES

46. SERIES
Final-121

47. SERIES Geometric Series: Geometric Series: Example
Write as a ratio of integers