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

What is the difference: SERIES What is the difference:

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

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. :

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

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

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SERIES Example:

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?

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

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SERIES Geometric Series: Geometric Series: prove:

SERIES Geometric Series: Geometric Series:

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SERIES Geometric Series: Geometric Series:

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)

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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.

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

SERIES THEOREM: Convergent THEOREM:THE TEST FOR DIVERGENCE Divergent

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

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

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

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

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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.

SERIES Reindexing Example We can write this geometric series

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

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

Extra Problems

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SERIES

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