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

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Presentation on theme: "SERIES DEF: A sequence is a list of numbers written in a definite order: DEF: Is called a series Example:"— Presentation transcript:

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

2 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

3 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. :

4 SERIES We define Given a series Given a series
Sequence of partial sums Given a series Sequence of partial sums

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

6 SERIES Example:

7 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?

8 SERIES Geometric Series: Geometric Series: Example Example Example:
Is it geometric?

9 SERIES Final-111

10 SERIES Final-102

11 SERIES Geometric Series: Geometric Series: prove:

12 SERIES Geometric Series: Geometric Series:

13 SERIES Final-102

14 SERIES Geometric Series: Geometric Series:

15 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)

16 SERIES Telescoping Series: Convergent Convergent Final-111

17 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.

18 SERIES Final-101 Final-112

19 SERIES Final-101

20 SERIES THEOREM: Convergent THEOREM:THE TEST FOR DIVERGENCE Divergent

21 THEOREM:THE TEST FOR DIVERGENCE
SERIES THEOREM:THE TEST FOR DIVERGENCE Divergent

22 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

23 SERIES THEOREM: Convergent REMARK(2): Seq. series REMARK(3): convg
Sequence REMARK(3): convg convg

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

25 SERIES Final-082

26 SERIES Final-081

27 SERIES Final-092

28 SERIES Final-121

29 SERIES Final-121

30 SERIES Final-103

31 SERIES

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

36 SERIES Geometric Series: Geometric Series: Example
Write as a ratio of integers

37

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49 SERIES Final-101

50 SERIES Final-112

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