(adsbygoogle = window.adsbygoogle || []).push({}); Series convergence "by Parts

Supose:

[tex] \sum c_n = \sum (a_n+b_n) [/tex] (*1)

[tex] \sum a_n [/tex] is conditionaly convergent (*2)

[tex] \sum b_n [/tex] is absolutly convergent (*3)

And I have seen this proof: [Proving [tex] \sum c_n[/tex] is conditionally convergent]

From (*1) and (*2) [tex]\Rightarrow[/tex] [tex] \sum c_n[/tex] its convergent [this one I understand, basic properties of series]

But now they do something like this: [proof by contradiction]

Supose [tex] \sum |c_n| [/tex] is convergent

so [tex] |a_n|\leq|c_n-b_n|\leq|c_n|+|b_n| [/tex] (How they "jump" to this conclusion?!)

and now the use comparison test to show that [tex]\sum c_n[/tex] is conditionally convergent. [No problems from here]

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# Series convergence by Parts

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