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Question: Cauchy criteria Power series

Question:Cauchy criteria Power series

Zeineb 530 Maple 17
series convergence
April 09 2022
1

Dear all

I have a power series, the genral term contains sign(sin(n)), 
everything is well defined, but when I apply summation to the general term, i can not see sign(sin(n)), its dropped.. why 

and how deduce the the limit to get the radius of convergence 

Cauchy_criteria.mw

thank you for any help 
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