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## SERIES

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**SERIES**DEF: A sequence is a list of numbers written in a definite order: DEF: Is called a series Example:**SERIES**DEF: Is called a series Example: its sum n-th term DEF: If the sum of the series convergent is finite number not infinity**SERIES**Example: DEF: Given a seris nth-partial sums : DEF: Given a seris the sequence of partial sums. :**SERIES**We define Given a series Sequence of partial sums Given a series Sequence of partial sums**SERIES**We define Given a series Sequence of partial sums DEF: If convergent convergent If divergent divergent**SERIES**Example:**SERIES**Special Series: Geometric Series: • Geometric Series • 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?**SERIES**Final-111**SERIES**Final-102**SERIES**Geometric Series: Geometric Series: prove:**SERIES**Geometric Series: Geometric Series:**SERIES**Final-102**SERIES**Geometric Series: Geometric Series:**SERIES**Special 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)**SERIES**Telescoping Series: Convergent Convergent Final-111**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.**SERIES**Final-101 Final-112**SERIES**Final-101**SERIES**THEOREM: Convergent THEOREM:THE TEST FOR DIVERGENCE Divergent**SERIES**THEOREM:THE TEST FOR DIVERGENCE 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 REMARK(2): Seq. series REMARK(3): Sequence convg convg**SERIES**REMARK Example All these items are true if these two series are convergent**SERIES**Final-082**SERIES**Final-081**SERIES**Final-092**SERIES**Final-121**SERIES**Final-121**SERIES**Final-103**SERIES**Adding or Deleting Terms Example REMARK(4): 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**SERIES THEOREM:THE TEST FOR DIVERGENCE Divergent convg convg**SERIES**Geometric Series: Geometric Series: Example Write as a ratio of integers**SERIES**Final-101**SERIES**Final-112