My book proves that if a series is absolutely convergent, then every rearangement is absolutely convergent also.(adsbygoogle = window.adsbygoogle || []).push({});

They then argue that the thm does not hold for conditionally convergeant series and give as a counter exemple the following thing: Let [itex]\sum a_n[/itex] be a conditionally convergent series and define thepositive partof {a_n} by p_n = a_n if a_n > 0 and =0 otherwise and thenegative partof {a_n} by q_n = a_n if a_n < 0 and =0 otherwise. Then observe that

[tex]p_n=\frac{a_n+|a_n|}{2}[/tex]

and

[tex]q_n=\frac{a_n-|a_n|}{2}[/tex]

or that inversely,

[tex]|a_n|=p_n-q_n[/tex] and [tex]a_n=p_n+q_n[/tex]

But what does this prove? Where is the rearangement?

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# Homework Help: Rearangement of series

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