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Lecture 24

Department

MathematicsCourse Code

MATH 2BProfessor

ERJAEE, G.Lecture

24This

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MATH 2B - Lecture 24 - Absolute Convergence / Ratio and Root Test

● Given a series

, we can consider the corresponding series

● A series

is called absolutely convergent if the series of absolute values

is convergent

Example:

absolutely convergent or not?

Solution:

● Consider

convergent (p=2)

● Therefore,

→

convergent absolutely convergent

Example:

absolutely convergent or not?

Solution:

● Consider

divergent (p=1)

●

is not absolutely convergent

● However, we know that the alternating series

is convergent

● So we can conclude that

is convergent, but not absolutely

convergent

Theorem: If a series

is absolutely convergent, then

must be

convergent

→

→

convergent absolutely convergent convergent

● “Absolutely convergent” is stronger than “convergent”

Example:

Determine whether

is convergent or not

Solution:

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